Research ArticleResearch on Geometric Mappings in Complex Systems Analysis

Yanyan Cui12 Chaojun Wang1 and Sifeng Zhu1

1College of Mathematics and Statistics Zhoukou Normal University Zhoukou Henan 466001 China2College of Mathematics and Information Science Hebei Normal University Shijiazhuang Hebei 050016 China

Correspondence should be addressed to Yanyan Cui cui9907081163com

Received 16 June 2016 Accepted 1 November 2016

Academic Editor Allan C Peterson

Copyright copy 2016 Yanyan Cui et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We mainly discuss the properties of a new subclass of starlike functions namely almost starlike functions of complex order 120582 inone and several complex variables We get the growth and distortion results for almost starlike functions of complex order 120582 By theproperties of functions with positive real parts and considering the zero of order 119896 we obtain the coefficient estimates for almoststarlike functions of complex order 120582 on 119863 We also discuss the invariance of almost starlike mappings of complex order 120582 onReinhardt domains and on the unit ball 119861 in complex Banach spaces The conclusions contain and generalize some known results

1 Introduction

The growth distortion theorems and coefficient estimatesfor univalent functions are important research contentsin geometric function theory of one and several complexvariables In one complex variable we have the followingknown growth and distortion theorems

Theorem 1 (see [1]) Let 119891 be a normalized biholomorphicfunction on the unit disk 119863 in C Then|119911|(1 + |119911|)2 le 1003816100381610038161003816119891 (119911)1003816100381610038161003816 le |119911|(1 minus |119911|)2 forall119911 isin 119863

1 minus |119911|(1 + |119911|)3 le 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 le 1 + |119911|(1 minus |119911|)3 forall119911 isin 119863 (1)

The distortion theorem for univalent functions on 119863was introduced by Koebe In the process of generalizing thedistortion theorem for univalent functions on 119863 to the unitball in C119899 Cartan et al [2 3] found the distortion theoremdid not hold for general biholomorphic mappings Cartan etal suggested restricting the mappings by geometric propertiessuch as starlikeness and convexity So many people began todiscuss the growth and distortion theorems for biholomorphicmappings with special geometric properties Starlike map-pings and convex mappings are discussed most [4ndash7] Whilethey discussed starlike mappings and convex mappings they

introduced some subclasses of those mappings Furthermorethey always discussed these new subclasses in one complexvariable firstlyThe growth and distortion results [8] for starlikefunctions are the same as Theorem 1 We have the followingresults with respect to convex functions

Theorem 2 (see [9]) Let 119891 be a convex function on 119863 with119891(0) = 0 and 1198911015840(0) = 1 Then|119911|1 + |119911| le 1003816100381610038161003816119891 (119911)1003816100381610038161003816 le |119911|1 minus |119911| forall119911 isin 1198631(1 + |119911|)2 le 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 le 1(1 minus |119911|)2 forall119911 isin 119863 (2)

In 1936 Robertson [10] obtained the growth coveringtheorems and coefficient estimates for starlike functions of order120572Theorem 3 (see [10]) Let 119891(119911) = 119911 + sum119899119895=2 119886119899119911119899 be a starlikefunction of order 120572 on119863 Then|119911|(1 + |119911|)2(1minus120572) le 1003816100381610038161003816119891 (119911)1003816100381610038161003816 le |119911|(1 minus |119911|)2(1minus120572) forall119911 isin 119863

10038161003816100381610038161198861198991003816100381610038161003816 le 1(119899 minus 1) 119899prod119896=2

(119896 minus 2120572) 119899 = 2 3 (3)

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2016 Article ID 4732704 11 pageshttpdxdoiorg10115520164732704

2 Discrete Dynamics in Nature and Society

In 1966 Boyd [11] obtained the coefficient estimates for thestarlike function 119891 of order 120572 where 120572 isin (0 1) and 119911 = 0 is thezero of order 119896+1 of119891(119911)minus119911 In 1975 Silverman [12] discussedthe distortion theorems and coefficient estimates for univalentanalytic functions with negative coefficients In 1991 Srivastavaand Owa [13] obtained the distortion theorems and coefficientestimates for a subclass 119878lowast(120572 120573 120574) of starlike functions

Recently there are many nice results about the growthdistortion theorems and coefficient estimates for subclassesof starlike functions and convex functions Obviously wehope there exist similar results in several complex variablesIn the process of discussing the properties of subclasses ofstarlike mappings and convex mappings we need the specificexamples of the mappings It is easy to find specific examplesof these new subclasses in C while it is very difficult in C119899

In 1995 Roper and Suffridge [14] introduced an operator

120601119899 (119891) (119911) = (119891 (1199111) radic1198911015840 (1199111)1199110)1015840 (4)

where 119911 = (1199111 1199110) isin 119861119899 1199111 isin 119863 1199110 = (1199112 119911119899) isin C119899minus1119891(1199111) isin 119867(119863) and radic1198911015840(0) = 1 They proved the operatorpreserves convexity and starlikeness on 119861119899 Graham andKohr [15] proved the Roper-Suffridge operator preserves theproperties of Bloch mappings on 119861119899 Thus we can constructlots of convex mappings and starlike mappings on 119861119899 by cor-responding functions on 119863 by the Roper-Suffridge operatorSo the Roper-Suffridge operator plays an important role inseveral complex variables Later many people generalized theRoper-Suffridge operator on different domains and differentspaces so as to construct biholomorphic mappings withspecific geometric properties in several complex variablesTherefore we can discuss the properties of biholomorphicmappings in several variables preferably In recent yearsthere are many results about the generalized Roper-Suffridgeoperators (such as [16ndash18])

In this paper we mainly discuss almost starlike functionsof complex order 120582 in one and several complex variables InSections 2 and 3 we discuss the growth distortion theoremsand coefficient estimates for almost starlike functions ofcomplex order 120582 on 119863 respectively In Section 4 we discussthe invariance of almost starlike mappings of complex order120582 on Reinhardt domains and on the unit ball 119861 in complexBanach spaces Thereby we can construct lots of almoststarlike mappings of complex order 120582 in several complexvariables through almost starlike functions of complex order120582 on119863 The conclusions generalize some known results

To get themain results we need the following definitions

Definition 4 (see [19]) Let Ω sub C119899 be a bounded starlikeand circular domain whose Minkowski functional 120588(119911) is1198621 except for a lower-dimensional manifold Let 119865(119911) be anormalized locally biholomorphic mapping onΩ LetR [(1 minus 120582) 2120588 (119911) 120597120588120597119911 (119911) 119869minus1119865 (119911) 119865 (119911)] ge minusR120582

119911 isin Ω 0 (5)

where 120582 isin C and Re 120582 le 0 Then 119865(119911) is called an almoststarlike mapping of complex order 120582 onΩ

For 119899 = 1 the condition in Definition 4 reduces to

R [(1 minus 120582) 119865 (119911)1199111198651015840 (119911)] ge minusR120582 (6)

Definition 5 (see [19]) Let 119865(119911) be a normalized locallybiholomorphicmapping on the unit ball119861 in complexBanachspaces Let

R (1 minus 120582) 119879119909 [(119863119865 (119909))minus1 119865 (119909)] ge minusR120582 119909 119909 isin 119861 0 (7)

where 120582 isin C and Re 120582 le 0 Then 119865(119911) is called an almoststarlike mapping of complex order 120582 on 119861

Setting 120582 = 120572(120572minus1) 120572 isin [0 1) in Definitions 4 and 5 weobtain the definition of almost starlike mappings of order 1205722 Growth and Distortion Results

Lemma 6 (see [20]) |(119911 minus 1199111)(119911 minus 1199112)| = 119896 (0 lt 119896 = 1 1199111 =1199112) indicates a circle in C whose center is 1199110 and whose radiusis 120588 where

1199110 = 1199111 minus 119896211991121 minus 1198962 120588 = 119896 10038161003816100381610038161199111 minus 1199112100381610038161003816100381610038161003816100381610038161 minus 11989621003816100381610038161003816 (8)

Theorem7 Let119891(119911) be an almost starlike function of complexorder 120582 on119863 with 120582 isin C minus1 Re 120582 le 0 Then

|119911| 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2 (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 minus |119911|)(|1+120582|minus1minusR120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 + |119911|)(minus|1+120582|minus1minusR120582)|1+120582|2 le 1003816100381610038161003816119891 (119911)1003816100381610038161003816 le |119911|sdot 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 minus |119911|)(minus|1+120582|minus1minusR120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 + |119911|)(|1+120582|minus1minusR120582)|1+120582|2

(9)

Proof Since 119891(119911) is an almost starlike function of complexorder 120582 on119863 then

R [(1 minus 120582) 119891 (119911)1199111198911015840 (119911)] ge R (minus120582) (10)

which follows1003816100381610038161003816100381610038161003816100381610038162R (minus120582) 1(1 minus 120582) (119891 (119911) 1199111198911015840 (119911)) minus 1100381610038161003816100381610038161003816100381610038161003816 le 1 (11)

Discrete Dynamics in Nature and Society 3

Let

119892 (119911) = 1199111198911015840 (119911)(1 minus 120582) 119891 (119911) 119911 isin 119863 0 11 minus 120582 119911 = 0 (12)

Therefore |2R(minus120582)119892(119911)minus1| le 1 Let119901(119911) = 2R(minus120582)119892(119911)minus1 We have |119901(119911)| le 1 and 119901(0) = 2R(minus120582)(1 minus 120582) minus 1 Letℎ (119911) = 119901 (119911) minus 119901 (0)1 minus 119901 (0)119901 (119911) 119911 isin 119863 (13)

Then ℎ(0) = 0 and |ℎ(119911)| lt 1 By Schwarz lemma we obtain|ℎ(119911)| le |119911| which follows10038161003816100381610038161003816100381610038161003816100381610038161003816 2R (minus120582) 119892 (119911) minus 2R (minus120582) (1 minus 120582)1 minus (2R (minus120582) (1 minus 120582) minus 1) [2R (minus120582) 119892 (119911) minus 1] 10038161003816100381610038161003816100381610038161003816100381610038161003816le |119911| (14)

For

1 minus (2R (minus120582)1 minus 120582 minus 1) [2R (minus120582) 119892 (119911) minus 1]= 1 + 1 + 1205821 minus 120582 [2R (minus120582) 119892 (119911) minus 1]= minus2R120582 [(1 + 120582) 119892 (119911) + 1]1 minus 120582

(15)

then by (14) we obtain10038161003816100381610038161003816100381610038161003816 (1 minus 120582) 119892 (119911) minus 1(1 + 120582) 119892 (119911) + 1 10038161003816100381610038161003816100381610038161003816 le |119911| (16)

which follows100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911) 119891 (119911) minus 11199111198911015840 (119911) 119891 (119911) + (1 minus 120582) (1 + 120582) 100381610038161003816100381610038161003816100381610038161003816 le 10038161003816100381610038161003816100381610038161003816 1 + 1205821 minus 12058210038161003816100381610038161003816100381610038161003816 |119911| (17)

By Lemma6we get that 1199111198911015840(119911)119891(119911) is in a circlewhose centeris 119886 and whose radius is 120588 where

119886 = 1 + |(1 + 120582) (1 minus 120582)|2 |119911|2 ((1 minus 120582) (1 + 120582))1 minus |(1 + 120582) (1 minus 120582)|2 |119911|2 120588 = |(1 + 120582) (1 minus 120582)| |119911| |1 + (1 minus 120582) (1 + 120582)|100381610038161003816100381610038161 minus |(1 + 120582) (1 minus 120582)|2 |119911|210038161003816100381610038161003816= 2 |119911| |1 minus 120582||1 minus 120582|2 minus |1 + 120582|2 |119911|2

(18)

So we obtain

R119886 minus 120588 le R1199111198911015840 (119911)119891 (119911) le R119886 + 120588 (19)

By direct computation we have

R119886 minus 120588 = 1 + |(1 + 120582) (1 minus 120582)|2 |119911|2R ((1 minus 120582) (1 + 120582))1 minus |(1 + 120582) (1 minus 120582)|2 |119911|2 minus 2 |119911| |1 minus 120582||1 minus 120582|2 minus |1 + 120582|2 |119911|2= 1 + |(1 + 120582) (1 minus 120582)|2 |119911|2 ((1 minus |120582|2) (1 + |120582|2 + 2R120582))1 minus |(1 + 120582) (1 minus 120582)|2 |119911|2 minus 2 |119911| |1 minus 120582||1 minus 120582|2 minus |1 + 120582|2 |119911|2= |1 + 120582|2 ((1 minus |120582|2) (1 + |120582|2 + 2R120582)) |119911|2 minus 2 |1 minus 120582| |119911| + |1 minus 120582|2|1 minus 120582|2 minus |1 + 120582|2 |119911|2

(20)

Similarly we can get

R119886 + 120588 = |1 + 120582|2 ((1 minus |120582|2) (1 + |120582|2 + 2R120582)) |119911|2 + 2 |1 minus 120582| |119911| + |1 minus 120582|2|1 minus 120582|2 minus |1 + 120582|2 |119911|2 = 119898 |119911|2 + 2 |1 minus 120582| |119911| + |1 minus 120582|2|1 minus 120582|2 minus |1 + 120582|2 |119911|2 (21)

where119898 = |1 + 120582|2((1 minus |120582|2)(1 + |120582|2 + 2R120582)) Let119891 (119909) = 1198981199092 + 2 |1 minus 120582| 119909 + |1 minus 120582|2|1 minus 120582|2 minus |1 + 120582|2 1199092 119909 isin [0 1) (22)

Immediately we have

1198911015840 (119909) = 119875(|1 minus 120582|2 minus |1 + 120582|2 1199092)2 (23)

4 Discrete Dynamics in Nature and Society

where119875 = (2119898119909 + 2 |1 minus 120582|) (|1 minus 120582|2 minus |1 + 120582|2 1199092) + (1198981199092+ 2 |1 minus 120582| 119909 + |1 minus 120582|2) 2 |1 + 120582|2 119909 = 2 |1 minus 120582| |1+ 120582|2 1199092 + 2 |1 minus 120582|2 (119898 + |1 + 120582|2) 119909 + 2 |1 minus 120582|3= 2 |1 minus 120582| [|1 + 120582|2 1199092 + |1 minus 120582| |1 + 120582|2sdot 2 (1 +R120582)1 + |120582|2 + 2R120582119909 + |1 minus 120582|2] = 2 |1 minus 120582| |1 + 120582|2sdot [[(119909 + |1 minus 120582| (1 +R120582)1 + |120582|2 + 2R120582 )2 + |1 minus 120582|2|1 + 120582|2minus |1 minus 120582|2 (1 +R120582)2(1 + |120582|2 + 2R120582)2]] = 2 |1 minus 120582| |1 + 120582|2(119909+ |1 minus 120582| (1 +R120582)1 + |120582|2 + 2R120582 )2+ |1 minus 120582|2|1 + 120582|2 (1 + |120582|2 + 2R120582)2 [(1 + |120582|2 + 2R120582)2minus |1 + 120582|2 (1 +R120582)2] = 2 |1 minus 120582| |1 + 120582|2 [[(119909+ |1 minus 120582| (1 +R120582)1 + |120582|2 + 2R120582 )2+ |1 minus 120582|2 (I120582)2|1 + 120582|2 (1 + |120582|2 + 2R120582)2]] ge 0

(24)

which follows 1198911015840(119909) ge 0 Therefore 119891(119909) is monotoneincreasing So

R119886 + 120588 = 119898 |119911|2 + 2 |1 minus 120582| |119911| + |1 minus 120582|2|1 minus 120582|2 minus |1 + 120582|2 |119911|2le 1198981199032 + 2 |1 minus 120582| 119903 + |1 minus 120582|2|1 minus 120582|2 minus |1 + 120582|2 1199032 (25)

where 119911 = 119903119890119894120579 120579 isin [0 2120587) By (19) we haveR(1199111198911015840(119911)119891(119911)) =119903(120597120597119903) ln |119891(119911)| le R119886 + 120588 Therefore120597120597119903 ln 1003816100381610038161003816119891 (119911)1003816100381610038161003816 le 1119903 1198981199032 + 2 |1 minus 120582| 119903 + |1 minus 120582|2|1 minus 120582|2 minus |1 + 120582|2 1199032= 1119903 (|1 minus 120582|2 minus |1 + 120582|2 1199032) [ |120582|2 minus 11 + |120582|2 + 2R120582 (|1minus 120582|2 minus |1 + 120582|2 1199032) + 1 minus |120582|21 + |120582|2 + 2R120582 |1 minus 120582|2 + |1

minus 120582|2 + 2 |1 minus 120582| 119903] = 1119903 [ |120582|2 minus 11 + |120582|2 + 2R120582 + 2 |1minus 120582| 119903 + (1 +R120582) |1 minus 120582| (1 + |120582|2 + 2R120582)|1 minus 120582|2 minus |1 + 120582|2 1199032 ]

(26)

Integrating both ends of (26) on [120576 119903] we obtainint119903120576

120597120597119905 ln 10038161003816100381610038161003816119891 (119905119890119894120579)10038161003816100381610038161003816 119889119905 le |120582|2 minus 11 + |120582|2 + 2R120582 int119903120576

119889119905119905 + 2 |1minus 120582| [int119903

120576

119889119905|1 minus 120582|2 minus |1 + 120582|2 1199052+ (1 +R120582) |1 minus 120582|1 + |120582|2 + 2R120582 int119903

120576

1119905 119889119905|1 minus 120582|2 minus |1 + 120582|2 1199052]= |120582|2 minus 11 + |120582|2 + 2R120582 ln 119905|119903120576 + 2 |1 minus 120582| 12 10038161003816100381610038161 minus 12058221003816100381610038161003816sdot ln 10038161003816100381610038161003816100381610038161003816 119905 + |(1 minus 120582) (1 + 120582)|119905 minus |(1 minus 120582) (1 + 120582)| 1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119903120576 + 2 (1 +R120582) |1 minus 120582|21 + |120582|2 + 2R120582sdot 12 |1 minus 120582|2 ln 119905210038161003816100381610038161003816|1 minus 120582|2 minus |1 + 120582|2 119905210038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816100381610038161003816119903

120576

(27)

Therefore

ln 1003816100381610038161003816119891 (119911)1003816100381610038161003816 minus ln 10038161003816100381610038161003816119891 (120576119890119894120579)10038161003816100381610038161003816 le |120582|2 minus 11 + |120582|2 + 2R120582 (ln 119903minus ln 120576) + 1|1 + 120582| (ln 10038161003816100381610038161003816100381610038161003816 119903 + |(1 minus 120582) (1 + 120582)|119903 minus |(1 minus 120582) (1 + 120582)| 10038161003816100381610038161003816100381610038161003816minus ln

10038161003816100381610038161003816100381610038161003816 120576 + |(1 minus 120582) (1 + 120582)|120576 minus |(1 minus 120582) (1 + 120582)| 10038161003816100381610038161003816100381610038161003816)+ (1 +R120582)1 + |120582|2 + 2R120582 (ln 119903210038161003816100381610038161003816|1 minus 120582|2 minus |1 + 120582|2 119903210038161003816100381610038161003816minus ln 120576210038161003816100381610038161003816|1 minus 120582|2 minus |1 + 120582|2 120576210038161003816100381610038161003816)

(28)

Since

ln 10038161003816100381610038161003816119891 (120576119890119894120579)10038161003816100381610038161003816 minus |120582|2 minus 11 + |120582|2 + 2R120582 ln 120576minus 1 +R1205821 + |120582|2 + 2R120582 ln 1205762 = ln

10038161003816100381610038161003816119891 (120576119890119894120579)10038161003816100381610038161003816120576 997888rarr 0(120576 997888rarr 0)

(29)

Discrete Dynamics in Nature and Society 5

setting 120576 rarr 0 in (28) we have

ln 1003816100381610038161003816119891 (119911)1003816100381610038161003816 le |120582|2 minus 11 + |120582|2 + 2R120582 ln 119903 + 1|1 + 120582|sdot ln 10038161003816100381610038161003816100381610038161003816 119903 + |(1 minus 120582) (1 + 120582)|119903 minus |(1 minus 120582) (1 + 120582)| 10038161003816100381610038161003816100381610038161003816+ 1 +R1205821 + |120582|2 + 2R120582 (ln 119903210038161003816100381610038161003816|1 minus 120582|2 minus |1 + 120582|2 119903210038161003816100381610038161003816+ ln |1 minus 120582|2)

(30)

Therefore1003816100381610038161003816119891 (119911)1003816100381610038161003816 le 119903(|120582|2minus1)(1+|120582|2+2R120582)sdot 10038161003816100381610038161003816100381610038161003816 119903 + |(1 minus 120582) (1 + 120582)|119903 minus |(1 minus 120582) (1 + 120582)| 100381610038161003816100381610038161003816100381610038161|1+120582|sdot ( |1 minus 120582|2 119903210038161003816100381610038161003816|1 minus 120582|2 minus |1 + 120582|2 119903210038161003816100381610038161003816)

(1+R120582)(1+|120582|2+2R120582)

= 119903 10038161003816100381610038161003816100381610038161003816 119903 + |(1 minus 120582) (1 + 120582)|119903 minus |(1 minus 120582) (1 + 120582)| 100381610038161003816100381610038161003816100381610038161|1+120582|sdot ( 1100381610038161003816100381610038161 minus |(1 + 120582) (1 minus 120582)|2 119903210038161003816100381610038161003816)

(1+R120582)(1+|120582|2+2R120582)

= 119903 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)(1+|120582|2+2R120582)sdot 10038161003816100381610038161003816100381610038161003816119903 minus 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 1205821003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816minus1|1+120582|minus(1+R120582)(1+|120582|2+2R120582)sdot 10038161003816100381610038161003816100381610038161003816119903 + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161|1+120582|minus(1+R120582)(1+|120582|2+2R120582)= 119903 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2sdot 10038161003816100381610038161003816100381610038161003816119903 minus 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 1205821003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(minus|1+120582|minus1minusR120582)|1+120582|2sdot 10038161003816100381610038161003816100381610038161003816119903 + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 1205821003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(|1+120582|minus1minusR120582)|1+120582|2 = |119911|sdot 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2sdot 10038161003816100381610038161003816100381610038161003816|119911| minus 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 1205821003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(minus|1+120582|minus1minusR120582)|1+120582|2sdot (|119911| + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816)(|1+120582|minus1minusR120582)|1+120582|2

(31)

Obviously |(1 minus 120582)(1 + 120582)| ge 1 forR120582 le 0 Thus |119911| le |(1 minus120582)(1 + 120582)| So we obtain1003816100381610038161003816119891 (119911)1003816100381610038161003816 le |119911| 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2

sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 minus |119911|)(minus|1+120582|minus1minusR120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 + |119911|)(|1+120582|minus1minusR120582)|1+120582|2

(32)

Similarly by (19) we can get

1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge |119911| 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 minus |119911|)(|1+120582|minus1minusR120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 + |119911|)(minus|1+120582|minus1minusR120582)|1+120582|2

(33)

Theorem8 Let119891(119911) be an almost starlike function of complexorder 120582 on119863 with 120582 = minus1 Then

1 minus |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le 1 + |119911| |119911| 119890minus|119911| le 1003816100381610038161003816119891 (119911)1003816100381610038161003816 le |119911| 119890|119911|(1 minus |119911|) 119890minus|119911| le 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 le (1 + |119911|) 119890|119911|

(34)

Proof By (14) we have |2119892(119911) minus 1| le |119911| which follows|1199111198911015840(119911)119891(119911) minus 1| le |119911| Therefore

1 minus |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le 1 + |119911| 1 minus |119911| le R

1199111198911015840 (119911)119891 (119911) le 1 + |119911| (35)

For |119911| = 119903 we have1 minus 119903 le 119903 120597120597119903 ln 1003816100381610038161003816119891 (119911)1003816100381610038161003816 = R

1199111198911015840 (119911)119891 (119911) le 1 + 119903 (36)

Similar to Theorem 7 we get

int119903120576

119889119905119905 minus int119903120576119889119905 le int119903

120576

120597120597119905 ln 10038161003816100381610038161003816119891 (119905119890119894120579)10038161003816100381610038161003816 119889119905le int119903120576

119889119905119905 + int119903120576119889119905 (37)

6 Discrete Dynamics in Nature and Society

Let 120576 rarr 0 we obtain |119911|119890minus|119911| le |119891(119911)| le |119911|119890|119911| Hence (1 minus|119911|)119890minus|119911| le |1198911015840(119911)| le (1 + |119911|)119890|119911|Setting 120582 = 120572(120572 minus 1) in Theorems 7 and 8 we get the

following corollary

Corollary 9 Let 119891(119911) be an almost starlike function of order120572 Then |119911|(1 + (1 minus 2120572) |119911|)2(1minus120572)(1minus2120572) le 1003816100381610038161003816119891 (119911)1003816100381610038161003816le |119911|(1 minus (1 minus 2120572) |119911|)2(1minus120572)(1minus2120572) 120572 isin [0 1) 12

|119911| 119890minus|119911| le 1003816100381610038161003816119891 (119911)1003816100381610038161003816 le |119911| 119890|119911| 120572 = 12 1 minus |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le 1 + |119911| (1 minus |119911|) 119890minus|119911| le 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 le (1 + |119911|) 119890|119911|

120572 = 12

(38)

Theorem 10 Let 119891(119911) be an almost starlike function ofcomplex order 120582 on119863 with 120582 isin C minus1 Re 120582 le 0 Then1 minus |119911|1 + |(1 + 120582) (1 minus 120582)| |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816le 1 + |119911|1 minus |(1 + 120582) (1 minus 120582)| |119911|

(1 minus |119911|) 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2+1sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 minus |119911|)(|1+120582|minus1minusR120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 + |119911|)(minus|1+120582|minus1minusR120582)|1+120582|2minus1 le 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816le (1 + |119911|) 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2+1sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 + |119911|)(|1+120582|minus1minusR120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 minus |119911|)(minus|1+120582|minus1minusR120582)|1+120582|2minus1

(39)

Proof By (17) we have100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) minus 1100381610038161003816100381610038161003816100381610038161003816 le 10038161003816100381610038161003816100381610038161003816 1 + 1205821 minus 12058210038161003816100381610038161003816100381610038161003816 |119911| 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) + 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038161003816 (40)

Since 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 minus 1 le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) minus 1100381610038161003816100381610038161003816100381610038161003816 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) + 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038161003816 le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 (41)

then we have100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 minus 1 le (100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816) 10038161003816100381610038161003816100381610038161003816 1 + 1205821 minus 12058210038161003816100381610038161003816100381610038161003816 |119911| (42)

which follows100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le 1 + |119911|1 minus |(1 + 120582) (1 minus 120582)| |119911| (43)

On the other hand we also have

1 minus 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le (100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816) 10038161003816100381610038161003816100381610038161003816 1 + 1205821 minus 12058210038161003816100381610038161003816100381610038161003816 |119911| (44)

which leads to100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 ge 1 minus |119911|1 + |(1 + 120582) (1 minus 120582)| |119911| (45)

Hence 1 minus |119911|1 + |(1 + 120582) (1 minus 120582)| |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816le 1 + |119911|1 minus |(1 + 120582) (1 minus 120582)| |119911| (46)

Then by Theorem 7 we obtain the desired conclusion (39)

Setting 120582 = 120572(120572 minus 1) inTheorem 10 we get the followingresult

Corollary 11 Let 119891(119911) be an almost starlike function of order120572 with 120572 isin [0 1) 12 Then1 minus |119911|1 + |1 minus 2120572| |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le 1 + |119911|1 minus |1 minus 2120572| |119911| 1 minus |119911|1 + |1 minus 2120572| |119911| |119911|(1 + (1 minus 2120572) |119911|)2(1minus120572)(1minus2120572)le 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816le 1 + |119911|1 minus |1 minus 2120572| |119911| |119911|(1 minus (1 minus 2120572) |119911|)2(1minus120572)(1minus2120572) (47)

3 Coefficient Estimates of Almost StarlikeFunctions of Complex Order 120582

Lemma 12 (see [21]) Let 119901(119911) = 1 + suminfin119899=1 119888119899119911119899 be aholomorphic function on 119863 with R119901(119911) gt 0 Then |119888119899| le2 (119899 = 1 2 )

Discrete Dynamics in Nature and Society 7

Lemma 13 (see [22]) Let119901(119911) = 1+suminfin119899=1 119888119899119911119899 be holomorphicon119863 ThenR119901(119911) gt 0 if and only if

119901 (119911) = 120585 minus 1120585 + 1 120585 isin 120597119863 (48)

Theorem 14 Let 119891(119911) = 119911 + suminfin119899=2 119886119899119911119899 be an almost starlikefunction of complex order 120582 with 120582 isin C Re 120582 le 0 on119863 Then10038161003816100381610038161198861198991003816100381610038161003816 le 119899minus1prod

119895=2

( 2|1 minus 120582| + 119895 minus 2119895 minus 1) 119899 = 2 3 (49)

Proof Since 119891(119911) is an almost starlike function of complexorder 120582 then

R [(1 minus 120582) 119891 (119911)1199111198911015840 (119911) + 120582] gt 0 119911 isin 119863 (50)

Let 119901(119911) = (1 minus 120582)(119891(119911)1199111198911015840(119911)) + 120582 Then we have1199111198911015840 (119911) (119901 (119911) minus 120582) = (1 minus 120582) 119891 (119911) (51)

and 119901(0) = 1 andR119901(119911) gt 0 Let 119901(119911) = 1 + suminfin119899=1 119888119899119911119899 Then10038161003816100381610038161198881198991003816100381610038161003816 le 2 (119899 = 1 2 ) (52)

by Lemma 12 For 119891(119911) = 119911 + suminfin119899=2 119886119899119911119899 (51) follows119911(1 + infinsum

119899=2

119899119886119899119911119899minus1)(1 minus 120582 + infinsum119899=1

119888119899119911119899)= (1 minus 120582)(119911 + infinsum

119899=2

119886119899119911119899) (53)

Therefore(1 minus 120582) 119911 + [1198881 + 21198862 (1 minus 120582)] 1199112 + sdot sdot sdot + [119888119899minus1+ 21198862119888119899minus2 + sdot sdot sdot + (119899 minus 1) 119886119899minus11198881 + 119899119886119899 (1 minus 120582)] 119911119899+ sdot sdot sdot = (1 minus 120582) 119911 + 1198862 (1 minus 120582) 1199112 + sdot sdot sdot + 119886119899 (1 minus 120582)sdot 119911119899 + sdot sdot sdot (54)

So 119886119899 = 119888119899minus1 + 21198862119888119899minus2 + sdot sdot sdot + (119899 minus 1) 119886119899minus11198881(119899 minus 1) (120582 minus 1) (55)

By (52) we get10038161003816100381610038161198861198991003816100381610038161003816 le 2 [1 + 2 100381610038161003816100381611988621003816100381610038161003816 + sdot sdot sdot + (119899 minus 1) 1003816100381610038161003816119886119899minus11003816100381610038161003816](119899 minus 1) |1 minus 120582| (56)

Bymathematical induction we obtain the desired conclusion

Theorem 15 Let 119891(119911) be an almost starlike function ofcomplex order 120582 with 120582 isin C Re 120582 le 0 on 119863 and let 119911 = 0be the zero of order 119896 + 1 (119896 isin N) of 119891(119911) minus 119911 Let 119891(119911) =119911 + suminfin119899=119896+1 119886119899119911119899 Then1003816100381610038161003816119886119899+11003816100381610038161003816 le 2|120582 + 119899| (119899 = 119896 119896 + 1 ) (57)

Proof Since 119891(119911) = 119911 + suminfin119899=119896+1 119886119899119911119899 similar to Theorem 14by (51) we obtain

119911(1 + infinsum119899=119896+1

119899119886119899119911119899minus1)(1 minus 120582 + infinsum119899=1

119888119899119911119899)= (1 minus 120582)(119911 + infinsum

119899=119896+1

119886119899119911119899) (58)

which follows(1 minus 120582) 119911 + 11988811199112 + sdot sdot sdot + 119888119896minus1119911119896+ [119888119896 + (119896 + 1) 119886119896+1] 119911119896+1 + sdot sdot sdot+ [119888119899 + (119899 + 1) 119886119899+1] 119911119899+1 + sdot sdot sdot= (1 minus 120582) 119911 + (1 minus 120582) 119886119896+1119911119896+1 + sdot sdot sdot+ (1 minus 120582) 119886119899+1119911119899+1 + sdot sdot sdot (59)

Therefore 1198881 = sdot sdot sdot = 119888119896minus1 = 0119888119899 = (minus120582 minus 119899) 119886119899+1 (119899 = 119896 119896 + 1 ) (60)

By (52) we get the desired conclusion

Theorem 16 119891(119911) = 119911 + suminfin119899=2 119886119899119911119899 is an almost starlikefunction of complex order 120582 with 120582 isin C and Re 120582 le 0 on 119863if and only ifinfinsum119899=2

[(1 + (119899 minus 1) 120582) (120585 + 1) + 119899 (1 minus 120585)] 119886119899119911119899 = minus2(120585 isin 120597119863 119899 = 2 3 ) (61)

Proof Let 119901(119911) = (1 minus 120582)(119891(119911)1199111198911015840(119911)) + 120582 Since 119891(119911) isan almost starlike function of complex order 120582 then 119901(119911)is holomorphic on 119863 and 119901(0) = 0 and R119901(119911) gt 0 ByLemma 13 we get(1 minus 120582) 119891 (119911)1199111198911015840 (119911) + 120582 = 120585 minus 1120585 + 1 120585 isin 120597119863 (62)

which follows(1 minus 120582) (120585 + 1) 119891 (119911) + 120582 (120585 + 1) 1199111198911015840 (119911)= (120585 minus 1) 1199111198911015840 (119911) (63)

For 119891(119911) = 119911 + suminfin119899=2 119886119899119911119899 we have(1 minus 120582) (120585 + 1) (119911 + infinsum119899=2

119886119899119911119899)+ 120582 (120585 + 1) (119911 + infinsum

119899=2

119899119886119899119911119899)= (120585 minus 1) (119911 + infinsum

119899=2

119899119886119899119911119899) (64)

which leads to the desired conclusion

8 Discrete Dynamics in Nature and Society

Remark 17 Setting 120582 = 120572(120572 minus 1) in Theorems 14ndash16 weget the corresponding results for almost starlike functions oforder 1205724 The Invariance of Almost Starlike Mappings

of Complex Order 120582In this section we mainly discuss the invariance of almoststarlike mappings of complex order 120582 on Reinhardt domainsand on the unit ball in complex Banach spaces under somegeneralized Roper-Suffridge operators

Lemma 18 (see [23]) Let Ω sub C119899 be a bounded andcompletely Reinhardt domain whose Minkowski functional120588(119911) is 1198621 except for a lower-dimensional manifold Ω0 Thenfor forall119911 = (1199111 119911119899) isin Ω Ω0 we have120597120588 (119911)120597119911 = (120597120588 (119911)1205971199111 120597120588 (119911)1205971199112 120597120588 (119911)120597119911119899 )

120597120588 (119911)120597119911119895 119911119895 ge 0120588 (119911) = 2 119899sum

119895=1

120597120588 (119911)120597119911119895 119911119895(65)

Theorem 19 Let 119891(119911) be an almost starlike function of com-plex order 120582 on 119863 with 120582 isin C and Re 120582 le 0 Let Ω sub C119899 be abounded and completely Reinhardt domain whose Minkowskifunctional 120588(119911) is 1198621 on Ω 0 Let

119865 (119911) = (119903119891(1199111119903 ) (119903119891 (1199111119903)1199111 )1205732sdot 1199112 (119903119891 (1199111119903)1199111 )120573119899 119911119899)1015840

(66)

where 120573119895 isin [0 1] 119903 = sup|1199111| 119911 = (1199111 119911119899)1015840 isin Ω and(119903119891(1199111119903)1199111)120573119895 |1199111=0 = 1 (119895 = 2 119899) Then 119865(119911) is an almoststarlike mapping of complex order 120582 on ΩProof For

119865 (119911) = (119903119891(1199111119903 ) (119903119891 (1199111119903)1199111 )1205732sdot 1199112 (119903119891 (1199111119903)1199111 )120573119899 119911119899)1015840

(67)

immediately we have

119869119865 (119911) =(((((((((((

1198911015840 (1199111119903 ) 0 sdot sdot sdot 01205732119903 [1198911015840 (1199111119903)119891 (1199111119903) minus 1199031199111 ](119903119891 (1199111119903)1199111 )1205732 1199112 (119903119891 (1199111119903)1199111 )1205732 sdot sdot sdot 0

sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot120573119899119903 [1198911015840 (1199111119903)119891 (1199111119903) minus 1199031199111](119903119891 (1199111119903)1199111 )120573119899 119911119899 0 sdot sdot sdot (119903119891 (1199111119903)1199111 )120573119899

)))))))))))

(68)

Therefore

119869minus1119865 (119911) =(((((((((((

11198911015840 (1199111119903) 0 sdot sdot sdot 01205732119903 [ 11990311991111198911015840 (1199111119903) minus 1119891 (1199111119903)] 1199112 (119903119891 (1199111119903)1199111 )minus1205732 sdot sdot sdot 0

sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot120573119899119903 [ 11990311991111198911015840 (1199111119903) minus 1119891 (1199111119903)] 119911119899 0 sdot sdot sdot (119903119891 (1199111119903)1199111 )minus120573119899

)))))))))))

(69)

Discrete Dynamics in Nature and Society 9

Then 119869minus1119865 (119911) 119865 (119911)= (119903119891 (1199111119903)1198911015840 (1199111119903) (1 minus 1205732 + 1205732 119903119891 (1199111119903)11991111198911015840 (1199111119903))sdot 1199112 (1 minus 120573119899 + 120573119899 119903119891 (1199111119903)11991111198911015840 (1199111119903)) 119911119899)1015840

(70)

By Lemma 18 we get(1 minus 120582) 2120597120588120597119911 (119911) 119869minus1119865 (119911) 119865 (119911) = 2 (1 minus 120582)sdot [[ 119903119891 (1199111119903)11991111198911015840 (1199111119903) 120597120588 (119911)1205971199111 1199111+ 119899sum119895=2

(1 minus 120573119895 + 120573119895 119903119891 (1199111119903)11991111198911015840 (1199111119903)) 120597120588 (119911)120597119911119895 119911119895]] (71)

Since119891(119911) is an almost starlike function of complex order120582 thenR[(1 minus 120582) 119903119891 (1199111119903)11991111198911015840 (1199111119903)] ge minusR120582 (72)

Therefore by Lemma 18 we obtain

R [(1 minus 120582) 2120597120588120597119911 (119911) 119869minus1119865 (119911) 119865 (119911)] = R[(1 minus 120582)sdot 119903119891 (1199111119903)11991111198911015840 (1199111119903)] 2120597120588 (119911)1205971199111 1199111 + 119899sum

119895=2

(1 minus 120573119895)R (1 minus 120582)+ 120573119895R[(1 minus 120582) 119903119891 (1199111119903)11991111198911015840 (1199111119903)] 2120597120588 (119911)120597119911119895 119911119895 ge minusR120582sdot 2120597120588 (119911)1205971199111 1199111 + 119899sum

119895=2

(1 minus 120573119895) (1 minusR120582) + 120573119895 (minusR120582)sdot 2120597120588 (119911)120597119911119895 119911119895 = (minusR120582) [2120597120588 (119911)1205971199111 1199111 + 2120597120588 (119911)120597119911119895 119911119895]+ 119899sum119895=2

(1 minus 120573119895) 2120597120588 (119911)120597119911119895 119911119895 ge (minusR120582) 120588 (119911)

(73)

Hence 119865(119911) is an almost starlike mapping of complex order 120582onΩ by Definition 4

Theorem 20 Let 119891119895 (119895 = 1 2 119899) be almost starlikefunctions of complex order 120582 on 119863 with 120582 isin C and Re 120582 lt 0Let Ω sub C119899 be a bounded starlike and circular domain whoseMinkowski functional 120588(119911) is1198621 onΩ0 Let 119903119895 be the radiusof the disk 119880119895 = 119911119895 isin C 119911 = (1199111 119911119895 119911119899)1015840 isin Ω whosecenter is zero Let

119865 (119911) = 119911 119899prod119895=1

(119903119895119891119895 (119911119895119903119895)119911119895 )120582119894 (74)

where 120582119894119895 ge 0 sum119899119895=1 120582119894119895 = 1 and (119891119895(119911119895)119911119895)120582119894119895 |119911119895=0 = 1 (119894 119895 =1 2 119899)Then119865(119911) is an almost starlikemapping of complexorder 120582 on ΩProof Let

120577119895 = 119911119895119903119895 119892 (119911) = 119899prod

119895=1

(119891119895 (120577119895)120577119895 )120582119894 (75)

which follows 119865(119911) = 119911119892(119911) Then 119869119865(119911) = 119869119892(119911)119911 + 119892(119911) Bydirect computation we get

119869119892 (119911) 119911 = 119892 (119911) [[119899sum119895=1

1205821198951198911015840119895 (119911119895119903119895) 119911119895119903119895119891119895 (119911119895119903119895) minus 1]]= 119892 (119911) [[

119899sum119895=1

1205821198951198911015840119895 (120577119895) 120577119895119891119895 (120577119895) minus 1]] (76)

Since 119891119895 are almost starlike functions of complex order 120582 on119863 then

R[[(1 minus 120582) 119891119895 (120577119895)1205771198951198911015840119895 (120577119895)]] ge minusR120582 gt 0 (77)

Therefore

R[ 11 minus 120582 (1 + 119869119892 (119911) 119911119892 (119911) )]= R[[ 11 minus 120582 119899sum

119895=1

1205821198951198911015840119895 (120577119895) 120577119895119891119895 (120577119895) ]]= 119899sum119895=1

120582119895R[[ 11 minus 120582 1198911015840119895 (120577119895) 120577119895119891119895 (120577119895) ]] gt 0(78)

Then 119869119892(119911)119911+119892(119911) = 0 It is obvious that119865(0) = 0 and 119869119865(0) =119868 so 119865(119911) is a normalized locally biholomorphic mapping onΩ and

119869minus1119865 (119911) 119865 (119911) = 119892 (119911) 119911119869119892 (119911) 119911 + 119892 (119911) (79)

Let

119902119895 (120577119895) = 2Re (minus120582) 1198911015840119895 (120577119895) 120577119895(1 minus 120582) 119891119895 (120577119895) minus 1 (80)

(77) follows 100381610038161003816100381610038161003816100381610038161003816100381610038162Re (minus120582) 1198911015840119895 (120577119895) 120577119895(1 minus 120582) 119891119895 (120577119895) minus 110038161003816100381610038161003816100381610038161003816100381610038161003816 le 1 (81)

10 Discrete Dynamics in Nature and Society

that is |119902119895(120577119895)| le 1 For Re 120582 lt 0 by (76) (79) and Lemma 18we obtain1003816100381610038161003816100381610038161003816100381610038162R (minus120582) 1(1 minus 120582) (2120588 (119911)) (120597120588120597119911) (119911) 119869minus1119865 (119911) 119865 (119911)

minus 1100381610038161003816100381610038161003816100381610038161003816 =100381610038161003816100381610038161003816100381610038161003816100381610038162R (minus120582) 11 minus 120582 119899sum

119895=1

1205821198951198911015840119895 (120577119895) 120577119895119891119895 (120577119895) minus 110038161003816100381610038161003816100381610038161003816100381610038161003816= 10038161003816100381610038161003816100381610038161003816100381610038161003816 119899sum119895=1120582119895119902119895 (120577119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816 le 119899sum119895=1120582119895 = 1(82)

which leads to

R [(1 minus 120582) 2120597120588120597119911 (119911) 119869minus1119865 (119911) 119865 (119911)] ge minusR120582120588 (119911) (83)

Hence 119865(119911) is an almost starlike mapping of complex order 120582onΩ by Definition 4

Similar to Theorem 20 we can get the following conclu-sion

Theorem 21 Let 119891119895 (119895 = 1 2 119899) be almost starlikefunctions of complex order 120582 on 119863 with 120582 isin C and Re 120582 lt 0Let

119865 (119909) = 119909 119899prod119895=1

(119891119895 (119879119906119895 (119909))119879119906119895 (119909) )120582119895 119909 isin 119861 (84)

where 120582119894119895 ge 0 sum119899119895=1 120582119894119895 = 1 (119891119895(119909119895)119909119895)120582119894119895 |119909119895=0 = 1 (119909119895 =119879119906119895(119909) 119894 119895 = 1 2 119899) and 119906119895 is the unit vector of thecomplex Banach space 119883 Then 119865(119909) is an almost starlikemapping of complex order 120582 on the unit ball 119861 in complexBanach space119883

Theorem 22 Let 119891 be an almost starlike function of complexorder 120582 on119863 with 120582 isin C and Re 120582 le 0 Let

119865 (119909) = 119899minus1sum119895=1

(119891(1198791199091 (119909))1198791199091 (119909) )120573119895 119879119909119895 (119909) 119909119895 + 119909minus 119899minus1sum119895=1

119879119909119895 (119909) 119909119895 119909 isin 119861 (85)

where 1199091 isin 119861 1199091 = 1 120573119895 isin [0 1] (119891(119911)119911)120573119895 |119911=0 = 1 (119895 =1 2 119899 minus 1) and 1199091 119909119899 are linearly independent and119879119909119894 isin 119883lowast such that 119879119909119894 = 1 119879119909119894(119909119894) = 1 and 119879119909119894(119909119895) =0 (119894 = 119895) for forall119909119894 Then 119865(119909) is an almost starlike mapping ofcomplex order 120582 on the unit ball 119861 in complex Banach space119883

Proof From [24] we get 119865(119909) is a normalized biholomorphicmapping on 119861 and119909 119879119909 [119863119865 (119909)minus1 119865 (119909)]= 1199092

+ 119899minus1sum119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162 [ 119891 (1198791199091 (119909))1198791199091 (119909) 1198911015840 (1198791199091 (119909)) minus 1] (86)

Since119891 is an almost starlike function of complex order120582 thenR[[(1 minus 120582) 119891 (119879119909119895 (119909))119879119909119895 (119909) 1198911015840 (119879119909119895 (119909))]] ge minusR120582 (87)

Therefore

R (1 minus 120582) 119879119909 [119863119865 (119909)minus1 119865 (119909)] = (1 minusR120582) 119909+ 119899minus1sum119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162119909sdotR(1 minus 120582) [ 119891 (1198791199091 (119909))1198791199091 (119909) 1198911015840 (1198791199091 (119909)) minus 1]ge (1 minusR120582) 119909 + 119899minus1sum

119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162119909sdot [R (minus120582) minus (1 minusR120582)] = minusR120582 119909+ 1119909 [[1199092 minus

119899minus1sum119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162]] ge minusR120582 119909

(88)

which lead to the desired conclusion by Definition 5

Similar to Theorem 22 we can get the following conclu-sion

Theorem 23 Let 119891 be an almost starlike function of complexorder 120582 on119863 with 120582 isin C Re 120582 le 0 Let119865 (119909) = Φ120573 (119891) (119909)= 119891 (1198791199091 (119909)) 1199091

+ (119891 (1198791199091 (119909))1198791199091 (119909) )120573 (119909 minus 1198791199091 (119909) 1199091) (89)

where 0 le 120573 le 1 1199091 isin 119861 1199091 = 1 (119891(1199111)1199111)120573|1199111=0 = 1 and119891(1199111) = 0 for 1199111 = 0 Then 119865(119909) is an almost starlike mappingof complex order 120582 on 119861Remark 24 Setting 120582 = 120572(120572 minus 1) in Theorems 19ndash23 weget the corresponding results with respect to almost starlikemappings of order 120572

Discrete Dynamics in Nature and Society 11

Competing Interests

The authors declare that they have no conflict of interests

Acknowledgments

This work is supported by NSF of China (no U1204618) andScience and Technology Research Projects of Henan Provin-cial Education Department (nos 14B110015 and 14B110016)

References

[1] C Pommerenke Univalent Functions Vandenhoeck amp Ru-precht Gottingen Germany 1975

[2] H Cartan ldquoSur la possibilite drsquoentendre aux functions de plus-ieurs variables complexes la theorie des functions univalentsrdquoin Lecons sur les Functions Univalents onMutivalents P MontelEd Gauthier-Villar Paris France 1933

[3] R W Barnard C H FitzGerald and S Gong ldquoA distortionthoerem for biholomorphicmappings inC2rdquoTransactions of theAmerican Mathematical Society vol 344 pp 902ndash924 1994

[4] T S Liu andW J Zhang ldquoA distortion thoerem for biholomor-phic convex mappings in C119899rdquo Chinese Journal of ContemporaryMathematics vol 20 no 3 pp 421ndash430 1999

[5] S Gong S K Wang and Q H Yu ldquoBiholomorphic convexmappings of ball in C119899rdquo Pacific Journal of Mathematics vol 161no 2 pp 287ndash306 1993

[6] S Gong and T S Liu ldquoDistortion theorems for biholomorphicconvex mappings on bounded convex circular domainsrdquo Chi-nese Annals of Mathematics Series B vol 20 no 3 pp 297ndash3041999

[7] I Graham and D Varolin ldquoBloch constants in one and severalvariablesrdquoPacific Journal ofMathematics vol 174 no 2 pp 347ndash357 1996

[8] P L DurenUnivalent Functions Springer New York NY USA1983

[9] T H Gronwall ldquoSome remarks on conformal representationrdquoAnnals of Mathematics vol 16 no 1ndash4 pp 72ndash76 191415

[10] M I Robertson ldquoOn the theory of univalent functionsrdquo Annalsof Mathematics vol 37 no 2 pp 374ndash408 1936

[11] A V Boyd ldquoCoefficient estimates for starlike functions of order120572rdquo Proceedings of the AmericanMathematical Society vol 17 pp1016ndash1018 1966

[12] H Silverman ldquoUnivalent functions with negative coefficientsrdquoProceedings of the American Mathematical Society vol 51 pp109ndash116 1975

[13] H M Srivastava and S Owa ldquoCertain subclasses of starlikefunctions Irdquo Journal ofMathematical Analysis and Applicationsvol 161 no 2 pp 405ndash415 1991

[14] K A Roper and T J Suffridge ldquoConvex mappings on the unitball ofC119899rdquo Journal drsquoAnalyseMathematique vol 65 pp 333ndash3471995

[15] I Graham and G Kohr ldquoUnivalent mappings associated withthe Roper-Suffridge extension operatorrdquo Journal drsquoAnalyseMathematique vol 81 pp 331ndash342 2000

[16] S Gong and T S Liu ldquoThe generalized Roper-Suffridge exten-sion operatorrdquo Journal of Mathematical Analysis and Applica-tions vol 284 no 2 pp 425ndash434 2003

[17] G Kohr ldquoLoewner chains and a modification of the Roper-Suffridge extension operatorrdquo Mathematica vol 71 no 1 pp41ndash48 2006

[18] J R Muir ldquoA class of Loewner chain preserving extensionoperatorsrdquo Journal of Mathematical Analysis and Applicationsvol 337 no 2 pp 862ndash879 2008

[19] Y H Zhao Almost Starlike Mappings of Complex Order 120582 onthe Unit Ball of a Complex Banach Space Zhejiang NormalUniversity Jinhua China 2013

[20] L V Ahlfors Complex Analysis McGraw-Hill New York NYUSA 3rd edition 1978

[21] L Bieberbach ldquoUber einige extremal problem in gebiete derkonformen abbildungrdquo Mathematische Annalen vol 77 no 2pp 153ndash172 1916

[22] T Hayami S Owa and H M Srivastava ldquoCoefficient inequal-ities for certain classes of analytic and univalent functionsrdquoJournal of Inequalities in Pure and Applied Mathematics vol 8no 4 pp 1ndash10 2007

[23] M S Liu and Y C Zhu ldquoGeneralized Roper-Suffridge oper-ators on bounded and complete Reinhardt domainsrdquo Sciencein China vol 37 no 10 pp 1193ndash1206 2007

[24] Y Y Cui and C J Wang ldquoThe generalized Roper-Suffridgeextension operators in complex Banach spacerdquo Journal ofMathematics vol 34 no 3 pp 515ndash520 2014

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Stochastic AnalysisInternational Journal of

2 Discrete Dynamics in Nature and Society

In 1966 Boyd [11] obtained the coefficient estimates for thestarlike function 119891 of order 120572 where 120572 isin (0 1) and 119911 = 0 is thezero of order 119896+1 of119891(119911)minus119911 In 1975 Silverman [12] discussedthe distortion theorems and coefficient estimates for univalentanalytic functions with negative coefficients In 1991 Srivastavaand Owa [13] obtained the distortion theorems and coefficientestimates for a subclass 119878lowast(120572 120573 120574) of starlike functions

Recently there are many nice results about the growthdistortion theorems and coefficient estimates for subclassesof starlike functions and convex functions Obviously wehope there exist similar results in several complex variablesIn the process of discussing the properties of subclasses ofstarlike mappings and convex mappings we need the specificexamples of the mappings It is easy to find specific examplesof these new subclasses in C while it is very difficult in C119899

In 1995 Roper and Suffridge [14] introduced an operator

120601119899 (119891) (119911) = (119891 (1199111) radic1198911015840 (1199111)1199110)1015840 (4)

where 119911 = (1199111 1199110) isin 119861119899 1199111 isin 119863 1199110 = (1199112 119911119899) isin C119899minus1119891(1199111) isin 119867(119863) and radic1198911015840(0) = 1 They proved the operatorpreserves convexity and starlikeness on 119861119899 Graham andKohr [15] proved the Roper-Suffridge operator preserves theproperties of Bloch mappings on 119861119899 Thus we can constructlots of convex mappings and starlike mappings on 119861119899 by cor-responding functions on 119863 by the Roper-Suffridge operatorSo the Roper-Suffridge operator plays an important role inseveral complex variables Later many people generalized theRoper-Suffridge operator on different domains and differentspaces so as to construct biholomorphic mappings withspecific geometric properties in several complex variablesTherefore we can discuss the properties of biholomorphicmappings in several variables preferably In recent yearsthere are many results about the generalized Roper-Suffridgeoperators (such as [16ndash18])

In this paper we mainly discuss almost starlike functionsof complex order 120582 in one and several complex variables InSections 2 and 3 we discuss the growth distortion theoremsand coefficient estimates for almost starlike functions ofcomplex order 120582 on 119863 respectively In Section 4 we discussthe invariance of almost starlike mappings of complex order120582 on Reinhardt domains and on the unit ball 119861 in complexBanach spaces Thereby we can construct lots of almoststarlike mappings of complex order 120582 in several complexvariables through almost starlike functions of complex order120582 on119863 The conclusions generalize some known results

To get themain results we need the following definitions

Definition 4 (see [19]) Let Ω sub C119899 be a bounded starlikeand circular domain whose Minkowski functional 120588(119911) is1198621 except for a lower-dimensional manifold Let 119865(119911) be anormalized locally biholomorphic mapping onΩ LetR [(1 minus 120582) 2120588 (119911) 120597120588120597119911 (119911) 119869minus1119865 (119911) 119865 (119911)] ge minusR120582

119911 isin Ω 0 (5)

where 120582 isin C and Re 120582 le 0 Then 119865(119911) is called an almoststarlike mapping of complex order 120582 onΩ

For 119899 = 1 the condition in Definition 4 reduces to

R [(1 minus 120582) 119865 (119911)1199111198651015840 (119911)] ge minusR120582 (6)

Definition 5 (see [19]) Let 119865(119911) be a normalized locallybiholomorphicmapping on the unit ball119861 in complexBanachspaces Let

R (1 minus 120582) 119879119909 [(119863119865 (119909))minus1 119865 (119909)] ge minusR120582 119909 119909 isin 119861 0 (7)

where 120582 isin C and Re 120582 le 0 Then 119865(119911) is called an almoststarlike mapping of complex order 120582 on 119861

Setting 120582 = 120572(120572minus1) 120572 isin [0 1) in Definitions 4 and 5 weobtain the definition of almost starlike mappings of order 1205722 Growth and Distortion Results

Lemma 6 (see [20]) |(119911 minus 1199111)(119911 minus 1199112)| = 119896 (0 lt 119896 = 1 1199111 =1199112) indicates a circle in C whose center is 1199110 and whose radiusis 120588 where

1199110 = 1199111 minus 119896211991121 minus 1198962 120588 = 119896 10038161003816100381610038161199111 minus 1199112100381610038161003816100381610038161003816100381610038161 minus 11989621003816100381610038161003816 (8)

Theorem7 Let119891(119911) be an almost starlike function of complexorder 120582 on119863 with 120582 isin C minus1 Re 120582 le 0 Then

|119911| 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2 (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 minus |119911|)(|1+120582|minus1minusR120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 + |119911|)(minus|1+120582|minus1minusR120582)|1+120582|2 le 1003816100381610038161003816119891 (119911)1003816100381610038161003816 le |119911|sdot 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 minus |119911|)(minus|1+120582|minus1minusR120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 + |119911|)(|1+120582|minus1minusR120582)|1+120582|2

(9)

Proof Since 119891(119911) is an almost starlike function of complexorder 120582 on119863 then

R [(1 minus 120582) 119891 (119911)1199111198911015840 (119911)] ge R (minus120582) (10)

which follows1003816100381610038161003816100381610038161003816100381610038162R (minus120582) 1(1 minus 120582) (119891 (119911) 1199111198911015840 (119911)) minus 1100381610038161003816100381610038161003816100381610038161003816 le 1 (11)

Discrete Dynamics in Nature and Society 3

Let

119892 (119911) = 1199111198911015840 (119911)(1 minus 120582) 119891 (119911) 119911 isin 119863 0 11 minus 120582 119911 = 0 (12)

Therefore |2R(minus120582)119892(119911)minus1| le 1 Let119901(119911) = 2R(minus120582)119892(119911)minus1 We have |119901(119911)| le 1 and 119901(0) = 2R(minus120582)(1 minus 120582) minus 1 Letℎ (119911) = 119901 (119911) minus 119901 (0)1 minus 119901 (0)119901 (119911) 119911 isin 119863 (13)

Then ℎ(0) = 0 and |ℎ(119911)| lt 1 By Schwarz lemma we obtain|ℎ(119911)| le |119911| which follows10038161003816100381610038161003816100381610038161003816100381610038161003816 2R (minus120582) 119892 (119911) minus 2R (minus120582) (1 minus 120582)1 minus (2R (minus120582) (1 minus 120582) minus 1) [2R (minus120582) 119892 (119911) minus 1] 10038161003816100381610038161003816100381610038161003816100381610038161003816le |119911| (14)

For

1 minus (2R (minus120582)1 minus 120582 minus 1) [2R (minus120582) 119892 (119911) minus 1]= 1 + 1 + 1205821 minus 120582 [2R (minus120582) 119892 (119911) minus 1]= minus2R120582 [(1 + 120582) 119892 (119911) + 1]1 minus 120582

(15)

then by (14) we obtain10038161003816100381610038161003816100381610038161003816 (1 minus 120582) 119892 (119911) minus 1(1 + 120582) 119892 (119911) + 1 10038161003816100381610038161003816100381610038161003816 le |119911| (16)

which follows100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911) 119891 (119911) minus 11199111198911015840 (119911) 119891 (119911) + (1 minus 120582) (1 + 120582) 100381610038161003816100381610038161003816100381610038161003816 le 10038161003816100381610038161003816100381610038161003816 1 + 1205821 minus 12058210038161003816100381610038161003816100381610038161003816 |119911| (17)

By Lemma6we get that 1199111198911015840(119911)119891(119911) is in a circlewhose centeris 119886 and whose radius is 120588 where

119886 = 1 + |(1 + 120582) (1 minus 120582)|2 |119911|2 ((1 minus 120582) (1 + 120582))1 minus |(1 + 120582) (1 minus 120582)|2 |119911|2 120588 = |(1 + 120582) (1 minus 120582)| |119911| |1 + (1 minus 120582) (1 + 120582)|100381610038161003816100381610038161 minus |(1 + 120582) (1 minus 120582)|2 |119911|210038161003816100381610038161003816= 2 |119911| |1 minus 120582||1 minus 120582|2 minus |1 + 120582|2 |119911|2

(18)

So we obtain

R119886 minus 120588 le R1199111198911015840 (119911)119891 (119911) le R119886 + 120588 (19)

By direct computation we have

R119886 minus 120588 = 1 + |(1 + 120582) (1 minus 120582)|2 |119911|2R ((1 minus 120582) (1 + 120582))1 minus |(1 + 120582) (1 minus 120582)|2 |119911|2 minus 2 |119911| |1 minus 120582||1 minus 120582|2 minus |1 + 120582|2 |119911|2= 1 + |(1 + 120582) (1 minus 120582)|2 |119911|2 ((1 minus |120582|2) (1 + |120582|2 + 2R120582))1 minus |(1 + 120582) (1 minus 120582)|2 |119911|2 minus 2 |119911| |1 minus 120582||1 minus 120582|2 minus |1 + 120582|2 |119911|2= |1 + 120582|2 ((1 minus |120582|2) (1 + |120582|2 + 2R120582)) |119911|2 minus 2 |1 minus 120582| |119911| + |1 minus 120582|2|1 minus 120582|2 minus |1 + 120582|2 |119911|2

(20)

Similarly we can get

R119886 + 120588 = |1 + 120582|2 ((1 minus |120582|2) (1 + |120582|2 + 2R120582)) |119911|2 + 2 |1 minus 120582| |119911| + |1 minus 120582|2|1 minus 120582|2 minus |1 + 120582|2 |119911|2 = 119898 |119911|2 + 2 |1 minus 120582| |119911| + |1 minus 120582|2|1 minus 120582|2 minus |1 + 120582|2 |119911|2 (21)

where119898 = |1 + 120582|2((1 minus |120582|2)(1 + |120582|2 + 2R120582)) Let119891 (119909) = 1198981199092 + 2 |1 minus 120582| 119909 + |1 minus 120582|2|1 minus 120582|2 minus |1 + 120582|2 1199092 119909 isin [0 1) (22)

Immediately we have

1198911015840 (119909) = 119875(|1 minus 120582|2 minus |1 + 120582|2 1199092)2 (23)

4 Discrete Dynamics in Nature and Society

where119875 = (2119898119909 + 2 |1 minus 120582|) (|1 minus 120582|2 minus |1 + 120582|2 1199092) + (1198981199092+ 2 |1 minus 120582| 119909 + |1 minus 120582|2) 2 |1 + 120582|2 119909 = 2 |1 minus 120582| |1+ 120582|2 1199092 + 2 |1 minus 120582|2 (119898 + |1 + 120582|2) 119909 + 2 |1 minus 120582|3= 2 |1 minus 120582| [|1 + 120582|2 1199092 + |1 minus 120582| |1 + 120582|2sdot 2 (1 +R120582)1 + |120582|2 + 2R120582119909 + |1 minus 120582|2] = 2 |1 minus 120582| |1 + 120582|2sdot [[(119909 + |1 minus 120582| (1 +R120582)1 + |120582|2 + 2R120582 )2 + |1 minus 120582|2|1 + 120582|2minus |1 minus 120582|2 (1 +R120582)2(1 + |120582|2 + 2R120582)2]] = 2 |1 minus 120582| |1 + 120582|2(119909+ |1 minus 120582| (1 +R120582)1 + |120582|2 + 2R120582 )2+ |1 minus 120582|2|1 + 120582|2 (1 + |120582|2 + 2R120582)2 [(1 + |120582|2 + 2R120582)2minus |1 + 120582|2 (1 +R120582)2] = 2 |1 minus 120582| |1 + 120582|2 [[(119909+ |1 minus 120582| (1 +R120582)1 + |120582|2 + 2R120582 )2+ |1 minus 120582|2 (I120582)2|1 + 120582|2 (1 + |120582|2 + 2R120582)2]] ge 0

(24)

which follows 1198911015840(119909) ge 0 Therefore 119891(119909) is monotoneincreasing So

R119886 + 120588 = 119898 |119911|2 + 2 |1 minus 120582| |119911| + |1 minus 120582|2|1 minus 120582|2 minus |1 + 120582|2 |119911|2le 1198981199032 + 2 |1 minus 120582| 119903 + |1 minus 120582|2|1 minus 120582|2 minus |1 + 120582|2 1199032 (25)

where 119911 = 119903119890119894120579 120579 isin [0 2120587) By (19) we haveR(1199111198911015840(119911)119891(119911)) =119903(120597120597119903) ln |119891(119911)| le R119886 + 120588 Therefore120597120597119903 ln 1003816100381610038161003816119891 (119911)1003816100381610038161003816 le 1119903 1198981199032 + 2 |1 minus 120582| 119903 + |1 minus 120582|2|1 minus 120582|2 minus |1 + 120582|2 1199032= 1119903 (|1 minus 120582|2 minus |1 + 120582|2 1199032) [ |120582|2 minus 11 + |120582|2 + 2R120582 (|1minus 120582|2 minus |1 + 120582|2 1199032) + 1 minus |120582|21 + |120582|2 + 2R120582 |1 minus 120582|2 + |1

minus 120582|2 + 2 |1 minus 120582| 119903] = 1119903 [ |120582|2 minus 11 + |120582|2 + 2R120582 + 2 |1minus 120582| 119903 + (1 +R120582) |1 minus 120582| (1 + |120582|2 + 2R120582)|1 minus 120582|2 minus |1 + 120582|2 1199032 ]

(26)

Integrating both ends of (26) on [120576 119903] we obtainint119903120576

120597120597119905 ln 10038161003816100381610038161003816119891 (119905119890119894120579)10038161003816100381610038161003816 119889119905 le |120582|2 minus 11 + |120582|2 + 2R120582 int119903120576

119889119905119905 + 2 |1minus 120582| [int119903

120576

119889119905|1 minus 120582|2 minus |1 + 120582|2 1199052+ (1 +R120582) |1 minus 120582|1 + |120582|2 + 2R120582 int119903

120576

1119905 119889119905|1 minus 120582|2 minus |1 + 120582|2 1199052]= |120582|2 minus 11 + |120582|2 + 2R120582 ln 119905|119903120576 + 2 |1 minus 120582| 12 10038161003816100381610038161 minus 12058221003816100381610038161003816sdot ln 10038161003816100381610038161003816100381610038161003816 119905 + |(1 minus 120582) (1 + 120582)|119905 minus |(1 minus 120582) (1 + 120582)| 1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119903120576 + 2 (1 +R120582) |1 minus 120582|21 + |120582|2 + 2R120582sdot 12 |1 minus 120582|2 ln 119905210038161003816100381610038161003816|1 minus 120582|2 minus |1 + 120582|2 119905210038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816100381610038161003816119903

120576

(27)

Therefore

ln 1003816100381610038161003816119891 (119911)1003816100381610038161003816 minus ln 10038161003816100381610038161003816119891 (120576119890119894120579)10038161003816100381610038161003816 le |120582|2 minus 11 + |120582|2 + 2R120582 (ln 119903minus ln 120576) + 1|1 + 120582| (ln 10038161003816100381610038161003816100381610038161003816 119903 + |(1 minus 120582) (1 + 120582)|119903 minus |(1 minus 120582) (1 + 120582)| 10038161003816100381610038161003816100381610038161003816minus ln

10038161003816100381610038161003816100381610038161003816 120576 + |(1 minus 120582) (1 + 120582)|120576 minus |(1 minus 120582) (1 + 120582)| 10038161003816100381610038161003816100381610038161003816)+ (1 +R120582)1 + |120582|2 + 2R120582 (ln 119903210038161003816100381610038161003816|1 minus 120582|2 minus |1 + 120582|2 119903210038161003816100381610038161003816minus ln 120576210038161003816100381610038161003816|1 minus 120582|2 minus |1 + 120582|2 120576210038161003816100381610038161003816)

(28)

Since

ln 10038161003816100381610038161003816119891 (120576119890119894120579)10038161003816100381610038161003816 minus |120582|2 minus 11 + |120582|2 + 2R120582 ln 120576minus 1 +R1205821 + |120582|2 + 2R120582 ln 1205762 = ln

10038161003816100381610038161003816119891 (120576119890119894120579)10038161003816100381610038161003816120576 997888rarr 0(120576 997888rarr 0)

(29)

Discrete Dynamics in Nature and Society 5

setting 120576 rarr 0 in (28) we have

ln 1003816100381610038161003816119891 (119911)1003816100381610038161003816 le |120582|2 minus 11 + |120582|2 + 2R120582 ln 119903 + 1|1 + 120582|sdot ln 10038161003816100381610038161003816100381610038161003816 119903 + |(1 minus 120582) (1 + 120582)|119903 minus |(1 minus 120582) (1 + 120582)| 10038161003816100381610038161003816100381610038161003816+ 1 +R1205821 + |120582|2 + 2R120582 (ln 119903210038161003816100381610038161003816|1 minus 120582|2 minus |1 + 120582|2 119903210038161003816100381610038161003816+ ln |1 minus 120582|2)

(30)

Therefore1003816100381610038161003816119891 (119911)1003816100381610038161003816 le 119903(|120582|2minus1)(1+|120582|2+2R120582)sdot 10038161003816100381610038161003816100381610038161003816 119903 + |(1 minus 120582) (1 + 120582)|119903 minus |(1 minus 120582) (1 + 120582)| 100381610038161003816100381610038161003816100381610038161|1+120582|sdot ( |1 minus 120582|2 119903210038161003816100381610038161003816|1 minus 120582|2 minus |1 + 120582|2 119903210038161003816100381610038161003816)

(1+R120582)(1+|120582|2+2R120582)

= 119903 10038161003816100381610038161003816100381610038161003816 119903 + |(1 minus 120582) (1 + 120582)|119903 minus |(1 minus 120582) (1 + 120582)| 100381610038161003816100381610038161003816100381610038161|1+120582|sdot ( 1100381610038161003816100381610038161 minus |(1 + 120582) (1 minus 120582)|2 119903210038161003816100381610038161003816)

(1+R120582)(1+|120582|2+2R120582)

= 119903 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)(1+|120582|2+2R120582)sdot 10038161003816100381610038161003816100381610038161003816119903 minus 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 1205821003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816minus1|1+120582|minus(1+R120582)(1+|120582|2+2R120582)sdot 10038161003816100381610038161003816100381610038161003816119903 + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161|1+120582|minus(1+R120582)(1+|120582|2+2R120582)= 119903 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2sdot 10038161003816100381610038161003816100381610038161003816119903 minus 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 1205821003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(minus|1+120582|minus1minusR120582)|1+120582|2sdot 10038161003816100381610038161003816100381610038161003816119903 + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 1205821003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(|1+120582|minus1minusR120582)|1+120582|2 = |119911|sdot 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2sdot 10038161003816100381610038161003816100381610038161003816|119911| minus 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 1205821003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(minus|1+120582|minus1minusR120582)|1+120582|2sdot (|119911| + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816)(|1+120582|minus1minusR120582)|1+120582|2

(31)

Obviously |(1 minus 120582)(1 + 120582)| ge 1 forR120582 le 0 Thus |119911| le |(1 minus120582)(1 + 120582)| So we obtain1003816100381610038161003816119891 (119911)1003816100381610038161003816 le |119911| 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2

sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 minus |119911|)(minus|1+120582|minus1minusR120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 + |119911|)(|1+120582|minus1minusR120582)|1+120582|2

(32)

Similarly by (19) we can get

1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge |119911| 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 minus |119911|)(|1+120582|minus1minusR120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 + |119911|)(minus|1+120582|minus1minusR120582)|1+120582|2

(33)

Theorem8 Let119891(119911) be an almost starlike function of complexorder 120582 on119863 with 120582 = minus1 Then

1 minus |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le 1 + |119911| |119911| 119890minus|119911| le 1003816100381610038161003816119891 (119911)1003816100381610038161003816 le |119911| 119890|119911|(1 minus |119911|) 119890minus|119911| le 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 le (1 + |119911|) 119890|119911|

(34)

Proof By (14) we have |2119892(119911) minus 1| le |119911| which follows|1199111198911015840(119911)119891(119911) minus 1| le |119911| Therefore

1 minus |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le 1 + |119911| 1 minus |119911| le R

1199111198911015840 (119911)119891 (119911) le 1 + |119911| (35)

For |119911| = 119903 we have1 minus 119903 le 119903 120597120597119903 ln 1003816100381610038161003816119891 (119911)1003816100381610038161003816 = R

1199111198911015840 (119911)119891 (119911) le 1 + 119903 (36)

Similar to Theorem 7 we get

int119903120576

119889119905119905 minus int119903120576119889119905 le int119903

120576

120597120597119905 ln 10038161003816100381610038161003816119891 (119905119890119894120579)10038161003816100381610038161003816 119889119905le int119903120576

119889119905119905 + int119903120576119889119905 (37)

6 Discrete Dynamics in Nature and Society

Let 120576 rarr 0 we obtain |119911|119890minus|119911| le |119891(119911)| le |119911|119890|119911| Hence (1 minus|119911|)119890minus|119911| le |1198911015840(119911)| le (1 + |119911|)119890|119911|Setting 120582 = 120572(120572 minus 1) in Theorems 7 and 8 we get the

following corollary

Corollary 9 Let 119891(119911) be an almost starlike function of order120572 Then |119911|(1 + (1 minus 2120572) |119911|)2(1minus120572)(1minus2120572) le 1003816100381610038161003816119891 (119911)1003816100381610038161003816le |119911|(1 minus (1 minus 2120572) |119911|)2(1minus120572)(1minus2120572) 120572 isin [0 1) 12

|119911| 119890minus|119911| le 1003816100381610038161003816119891 (119911)1003816100381610038161003816 le |119911| 119890|119911| 120572 = 12 1 minus |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le 1 + |119911| (1 minus |119911|) 119890minus|119911| le 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 le (1 + |119911|) 119890|119911|

120572 = 12

(38)

Theorem 10 Let 119891(119911) be an almost starlike function ofcomplex order 120582 on119863 with 120582 isin C minus1 Re 120582 le 0 Then1 minus |119911|1 + |(1 + 120582) (1 minus 120582)| |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816le 1 + |119911|1 minus |(1 + 120582) (1 minus 120582)| |119911|

(1 minus |119911|) 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2+1sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 minus |119911|)(|1+120582|minus1minusR120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 + |119911|)(minus|1+120582|minus1minusR120582)|1+120582|2minus1 le 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816le (1 + |119911|) 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2+1sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 + |119911|)(|1+120582|minus1minusR120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 minus |119911|)(minus|1+120582|minus1minusR120582)|1+120582|2minus1

(39)

Proof By (17) we have100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) minus 1100381610038161003816100381610038161003816100381610038161003816 le 10038161003816100381610038161003816100381610038161003816 1 + 1205821 minus 12058210038161003816100381610038161003816100381610038161003816 |119911| 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) + 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038161003816 (40)

Since 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 minus 1 le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) minus 1100381610038161003816100381610038161003816100381610038161003816 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) + 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038161003816 le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 (41)

then we have100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 minus 1 le (100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816) 10038161003816100381610038161003816100381610038161003816 1 + 1205821 minus 12058210038161003816100381610038161003816100381610038161003816 |119911| (42)

which follows100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le 1 + |119911|1 minus |(1 + 120582) (1 minus 120582)| |119911| (43)

On the other hand we also have

1 minus 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le (100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816) 10038161003816100381610038161003816100381610038161003816 1 + 1205821 minus 12058210038161003816100381610038161003816100381610038161003816 |119911| (44)

which leads to100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 ge 1 minus |119911|1 + |(1 + 120582) (1 minus 120582)| |119911| (45)

Hence 1 minus |119911|1 + |(1 + 120582) (1 minus 120582)| |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816le 1 + |119911|1 minus |(1 + 120582) (1 minus 120582)| |119911| (46)

Then by Theorem 7 we obtain the desired conclusion (39)

Setting 120582 = 120572(120572 minus 1) inTheorem 10 we get the followingresult

Corollary 11 Let 119891(119911) be an almost starlike function of order120572 with 120572 isin [0 1) 12 Then1 minus |119911|1 + |1 minus 2120572| |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le 1 + |119911|1 minus |1 minus 2120572| |119911| 1 minus |119911|1 + |1 minus 2120572| |119911| |119911|(1 + (1 minus 2120572) |119911|)2(1minus120572)(1minus2120572)le 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816le 1 + |119911|1 minus |1 minus 2120572| |119911| |119911|(1 minus (1 minus 2120572) |119911|)2(1minus120572)(1minus2120572) (47)

3 Coefficient Estimates of Almost StarlikeFunctions of Complex Order 120582

Lemma 12 (see [21]) Let 119901(119911) = 1 + suminfin119899=1 119888119899119911119899 be aholomorphic function on 119863 with R119901(119911) gt 0 Then |119888119899| le2 (119899 = 1 2 )

Discrete Dynamics in Nature and Society 7

Lemma 13 (see [22]) Let119901(119911) = 1+suminfin119899=1 119888119899119911119899 be holomorphicon119863 ThenR119901(119911) gt 0 if and only if

119901 (119911) = 120585 minus 1120585 + 1 120585 isin 120597119863 (48)

Theorem 14 Let 119891(119911) = 119911 + suminfin119899=2 119886119899119911119899 be an almost starlikefunction of complex order 120582 with 120582 isin C Re 120582 le 0 on119863 Then10038161003816100381610038161198861198991003816100381610038161003816 le 119899minus1prod

119895=2

( 2|1 minus 120582| + 119895 minus 2119895 minus 1) 119899 = 2 3 (49)

Proof Since 119891(119911) is an almost starlike function of complexorder 120582 then

R [(1 minus 120582) 119891 (119911)1199111198911015840 (119911) + 120582] gt 0 119911 isin 119863 (50)

Let 119901(119911) = (1 minus 120582)(119891(119911)1199111198911015840(119911)) + 120582 Then we have1199111198911015840 (119911) (119901 (119911) minus 120582) = (1 minus 120582) 119891 (119911) (51)

and 119901(0) = 1 andR119901(119911) gt 0 Let 119901(119911) = 1 + suminfin119899=1 119888119899119911119899 Then10038161003816100381610038161198881198991003816100381610038161003816 le 2 (119899 = 1 2 ) (52)

by Lemma 12 For 119891(119911) = 119911 + suminfin119899=2 119886119899119911119899 (51) follows119911(1 + infinsum

119899=2

119899119886119899119911119899minus1)(1 minus 120582 + infinsum119899=1

119888119899119911119899)= (1 minus 120582)(119911 + infinsum

119899=2

119886119899119911119899) (53)

Therefore(1 minus 120582) 119911 + [1198881 + 21198862 (1 minus 120582)] 1199112 + sdot sdot sdot + [119888119899minus1+ 21198862119888119899minus2 + sdot sdot sdot + (119899 minus 1) 119886119899minus11198881 + 119899119886119899 (1 minus 120582)] 119911119899+ sdot sdot sdot = (1 minus 120582) 119911 + 1198862 (1 minus 120582) 1199112 + sdot sdot sdot + 119886119899 (1 minus 120582)sdot 119911119899 + sdot sdot sdot (54)

So 119886119899 = 119888119899minus1 + 21198862119888119899minus2 + sdot sdot sdot + (119899 minus 1) 119886119899minus11198881(119899 minus 1) (120582 minus 1) (55)

By (52) we get10038161003816100381610038161198861198991003816100381610038161003816 le 2 [1 + 2 100381610038161003816100381611988621003816100381610038161003816 + sdot sdot sdot + (119899 minus 1) 1003816100381610038161003816119886119899minus11003816100381610038161003816](119899 minus 1) |1 minus 120582| (56)

Bymathematical induction we obtain the desired conclusion

Theorem 15 Let 119891(119911) be an almost starlike function ofcomplex order 120582 with 120582 isin C Re 120582 le 0 on 119863 and let 119911 = 0be the zero of order 119896 + 1 (119896 isin N) of 119891(119911) minus 119911 Let 119891(119911) =119911 + suminfin119899=119896+1 119886119899119911119899 Then1003816100381610038161003816119886119899+11003816100381610038161003816 le 2|120582 + 119899| (119899 = 119896 119896 + 1 ) (57)

Proof Since 119891(119911) = 119911 + suminfin119899=119896+1 119886119899119911119899 similar to Theorem 14by (51) we obtain

119911(1 + infinsum119899=119896+1

119899119886119899119911119899minus1)(1 minus 120582 + infinsum119899=1

119888119899119911119899)= (1 minus 120582)(119911 + infinsum

119899=119896+1

119886119899119911119899) (58)

which follows(1 minus 120582) 119911 + 11988811199112 + sdot sdot sdot + 119888119896minus1119911119896+ [119888119896 + (119896 + 1) 119886119896+1] 119911119896+1 + sdot sdot sdot+ [119888119899 + (119899 + 1) 119886119899+1] 119911119899+1 + sdot sdot sdot= (1 minus 120582) 119911 + (1 minus 120582) 119886119896+1119911119896+1 + sdot sdot sdot+ (1 minus 120582) 119886119899+1119911119899+1 + sdot sdot sdot (59)

Therefore 1198881 = sdot sdot sdot = 119888119896minus1 = 0119888119899 = (minus120582 minus 119899) 119886119899+1 (119899 = 119896 119896 + 1 ) (60)

By (52) we get the desired conclusion

Theorem 16 119891(119911) = 119911 + suminfin119899=2 119886119899119911119899 is an almost starlikefunction of complex order 120582 with 120582 isin C and Re 120582 le 0 on 119863if and only ifinfinsum119899=2

[(1 + (119899 minus 1) 120582) (120585 + 1) + 119899 (1 minus 120585)] 119886119899119911119899 = minus2(120585 isin 120597119863 119899 = 2 3 ) (61)

Proof Let 119901(119911) = (1 minus 120582)(119891(119911)1199111198911015840(119911)) + 120582 Since 119891(119911) isan almost starlike function of complex order 120582 then 119901(119911)is holomorphic on 119863 and 119901(0) = 0 and R119901(119911) gt 0 ByLemma 13 we get(1 minus 120582) 119891 (119911)1199111198911015840 (119911) + 120582 = 120585 minus 1120585 + 1 120585 isin 120597119863 (62)

which follows(1 minus 120582) (120585 + 1) 119891 (119911) + 120582 (120585 + 1) 1199111198911015840 (119911)= (120585 minus 1) 1199111198911015840 (119911) (63)

For 119891(119911) = 119911 + suminfin119899=2 119886119899119911119899 we have(1 minus 120582) (120585 + 1) (119911 + infinsum119899=2

119886119899119911119899)+ 120582 (120585 + 1) (119911 + infinsum

119899=2

119899119886119899119911119899)= (120585 minus 1) (119911 + infinsum

119899=2

119899119886119899119911119899) (64)

which leads to the desired conclusion

8 Discrete Dynamics in Nature and Society

Remark 17 Setting 120582 = 120572(120572 minus 1) in Theorems 14ndash16 weget the corresponding results for almost starlike functions oforder 1205724 The Invariance of Almost Starlike Mappings

of Complex Order 120582In this section we mainly discuss the invariance of almoststarlike mappings of complex order 120582 on Reinhardt domainsand on the unit ball in complex Banach spaces under somegeneralized Roper-Suffridge operators

Lemma 18 (see [23]) Let Ω sub C119899 be a bounded andcompletely Reinhardt domain whose Minkowski functional120588(119911) is 1198621 except for a lower-dimensional manifold Ω0 Thenfor forall119911 = (1199111 119911119899) isin Ω Ω0 we have120597120588 (119911)120597119911 = (120597120588 (119911)1205971199111 120597120588 (119911)1205971199112 120597120588 (119911)120597119911119899 )

120597120588 (119911)120597119911119895 119911119895 ge 0120588 (119911) = 2 119899sum

119895=1

120597120588 (119911)120597119911119895 119911119895(65)

Theorem 19 Let 119891(119911) be an almost starlike function of com-plex order 120582 on 119863 with 120582 isin C and Re 120582 le 0 Let Ω sub C119899 be abounded and completely Reinhardt domain whose Minkowskifunctional 120588(119911) is 1198621 on Ω 0 Let

119865 (119911) = (119903119891(1199111119903 ) (119903119891 (1199111119903)1199111 )1205732sdot 1199112 (119903119891 (1199111119903)1199111 )120573119899 119911119899)1015840

(66)

where 120573119895 isin [0 1] 119903 = sup|1199111| 119911 = (1199111 119911119899)1015840 isin Ω and(119903119891(1199111119903)1199111)120573119895 |1199111=0 = 1 (119895 = 2 119899) Then 119865(119911) is an almoststarlike mapping of complex order 120582 on ΩProof For

119865 (119911) = (119903119891(1199111119903 ) (119903119891 (1199111119903)1199111 )1205732sdot 1199112 (119903119891 (1199111119903)1199111 )120573119899 119911119899)1015840

(67)

immediately we have

119869119865 (119911) =(((((((((((

1198911015840 (1199111119903 ) 0 sdot sdot sdot 01205732119903 [1198911015840 (1199111119903)119891 (1199111119903) minus 1199031199111 ](119903119891 (1199111119903)1199111 )1205732 1199112 (119903119891 (1199111119903)1199111 )1205732 sdot sdot sdot 0

sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot120573119899119903 [1198911015840 (1199111119903)119891 (1199111119903) minus 1199031199111](119903119891 (1199111119903)1199111 )120573119899 119911119899 0 sdot sdot sdot (119903119891 (1199111119903)1199111 )120573119899

)))))))))))

(68)

Therefore

119869minus1119865 (119911) =(((((((((((

11198911015840 (1199111119903) 0 sdot sdot sdot 01205732119903 [ 11990311991111198911015840 (1199111119903) minus 1119891 (1199111119903)] 1199112 (119903119891 (1199111119903)1199111 )minus1205732 sdot sdot sdot 0

sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot120573119899119903 [ 11990311991111198911015840 (1199111119903) minus 1119891 (1199111119903)] 119911119899 0 sdot sdot sdot (119903119891 (1199111119903)1199111 )minus120573119899

)))))))))))

(69)

Discrete Dynamics in Nature and Society 9

Then 119869minus1119865 (119911) 119865 (119911)= (119903119891 (1199111119903)1198911015840 (1199111119903) (1 minus 1205732 + 1205732 119903119891 (1199111119903)11991111198911015840 (1199111119903))sdot 1199112 (1 minus 120573119899 + 120573119899 119903119891 (1199111119903)11991111198911015840 (1199111119903)) 119911119899)1015840

(70)

By Lemma 18 we get(1 minus 120582) 2120597120588120597119911 (119911) 119869minus1119865 (119911) 119865 (119911) = 2 (1 minus 120582)sdot [[ 119903119891 (1199111119903)11991111198911015840 (1199111119903) 120597120588 (119911)1205971199111 1199111+ 119899sum119895=2

(1 minus 120573119895 + 120573119895 119903119891 (1199111119903)11991111198911015840 (1199111119903)) 120597120588 (119911)120597119911119895 119911119895]] (71)

Since119891(119911) is an almost starlike function of complex order120582 thenR[(1 minus 120582) 119903119891 (1199111119903)11991111198911015840 (1199111119903)] ge minusR120582 (72)

Therefore by Lemma 18 we obtain

R [(1 minus 120582) 2120597120588120597119911 (119911) 119869minus1119865 (119911) 119865 (119911)] = R[(1 minus 120582)sdot 119903119891 (1199111119903)11991111198911015840 (1199111119903)] 2120597120588 (119911)1205971199111 1199111 + 119899sum

119895=2

(1 minus 120573119895)R (1 minus 120582)+ 120573119895R[(1 minus 120582) 119903119891 (1199111119903)11991111198911015840 (1199111119903)] 2120597120588 (119911)120597119911119895 119911119895 ge minusR120582sdot 2120597120588 (119911)1205971199111 1199111 + 119899sum

119895=2

(1 minus 120573119895) (1 minusR120582) + 120573119895 (minusR120582)sdot 2120597120588 (119911)120597119911119895 119911119895 = (minusR120582) [2120597120588 (119911)1205971199111 1199111 + 2120597120588 (119911)120597119911119895 119911119895]+ 119899sum119895=2

(1 minus 120573119895) 2120597120588 (119911)120597119911119895 119911119895 ge (minusR120582) 120588 (119911)

(73)

Hence 119865(119911) is an almost starlike mapping of complex order 120582onΩ by Definition 4

Theorem 20 Let 119891119895 (119895 = 1 2 119899) be almost starlikefunctions of complex order 120582 on 119863 with 120582 isin C and Re 120582 lt 0Let Ω sub C119899 be a bounded starlike and circular domain whoseMinkowski functional 120588(119911) is1198621 onΩ0 Let 119903119895 be the radiusof the disk 119880119895 = 119911119895 isin C 119911 = (1199111 119911119895 119911119899)1015840 isin Ω whosecenter is zero Let

119865 (119911) = 119911 119899prod119895=1

(119903119895119891119895 (119911119895119903119895)119911119895 )120582119894 (74)

where 120582119894119895 ge 0 sum119899119895=1 120582119894119895 = 1 and (119891119895(119911119895)119911119895)120582119894119895 |119911119895=0 = 1 (119894 119895 =1 2 119899)Then119865(119911) is an almost starlikemapping of complexorder 120582 on ΩProof Let

120577119895 = 119911119895119903119895 119892 (119911) = 119899prod

119895=1

(119891119895 (120577119895)120577119895 )120582119894 (75)

which follows 119865(119911) = 119911119892(119911) Then 119869119865(119911) = 119869119892(119911)119911 + 119892(119911) Bydirect computation we get

119869119892 (119911) 119911 = 119892 (119911) [[119899sum119895=1

1205821198951198911015840119895 (119911119895119903119895) 119911119895119903119895119891119895 (119911119895119903119895) minus 1]]= 119892 (119911) [[

119899sum119895=1

1205821198951198911015840119895 (120577119895) 120577119895119891119895 (120577119895) minus 1]] (76)

Since 119891119895 are almost starlike functions of complex order 120582 on119863 then

R[[(1 minus 120582) 119891119895 (120577119895)1205771198951198911015840119895 (120577119895)]] ge minusR120582 gt 0 (77)

Therefore

R[ 11 minus 120582 (1 + 119869119892 (119911) 119911119892 (119911) )]= R[[ 11 minus 120582 119899sum

119895=1

1205821198951198911015840119895 (120577119895) 120577119895119891119895 (120577119895) ]]= 119899sum119895=1

120582119895R[[ 11 minus 120582 1198911015840119895 (120577119895) 120577119895119891119895 (120577119895) ]] gt 0(78)

Then 119869119892(119911)119911+119892(119911) = 0 It is obvious that119865(0) = 0 and 119869119865(0) =119868 so 119865(119911) is a normalized locally biholomorphic mapping onΩ and

119869minus1119865 (119911) 119865 (119911) = 119892 (119911) 119911119869119892 (119911) 119911 + 119892 (119911) (79)

Let

119902119895 (120577119895) = 2Re (minus120582) 1198911015840119895 (120577119895) 120577119895(1 minus 120582) 119891119895 (120577119895) minus 1 (80)

(77) follows 100381610038161003816100381610038161003816100381610038161003816100381610038162Re (minus120582) 1198911015840119895 (120577119895) 120577119895(1 minus 120582) 119891119895 (120577119895) minus 110038161003816100381610038161003816100381610038161003816100381610038161003816 le 1 (81)

10 Discrete Dynamics in Nature and Society

that is |119902119895(120577119895)| le 1 For Re 120582 lt 0 by (76) (79) and Lemma 18we obtain1003816100381610038161003816100381610038161003816100381610038162R (minus120582) 1(1 minus 120582) (2120588 (119911)) (120597120588120597119911) (119911) 119869minus1119865 (119911) 119865 (119911)

minus 1100381610038161003816100381610038161003816100381610038161003816 =100381610038161003816100381610038161003816100381610038161003816100381610038162R (minus120582) 11 minus 120582 119899sum

119895=1

1205821198951198911015840119895 (120577119895) 120577119895119891119895 (120577119895) minus 110038161003816100381610038161003816100381610038161003816100381610038161003816= 10038161003816100381610038161003816100381610038161003816100381610038161003816 119899sum119895=1120582119895119902119895 (120577119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816 le 119899sum119895=1120582119895 = 1(82)

which leads to

R [(1 minus 120582) 2120597120588120597119911 (119911) 119869minus1119865 (119911) 119865 (119911)] ge minusR120582120588 (119911) (83)

Hence 119865(119911) is an almost starlike mapping of complex order 120582onΩ by Definition 4

Similar to Theorem 20 we can get the following conclu-sion

Theorem 21 Let 119891119895 (119895 = 1 2 119899) be almost starlikefunctions of complex order 120582 on 119863 with 120582 isin C and Re 120582 lt 0Let

119865 (119909) = 119909 119899prod119895=1

(119891119895 (119879119906119895 (119909))119879119906119895 (119909) )120582119895 119909 isin 119861 (84)

where 120582119894119895 ge 0 sum119899119895=1 120582119894119895 = 1 (119891119895(119909119895)119909119895)120582119894119895 |119909119895=0 = 1 (119909119895 =119879119906119895(119909) 119894 119895 = 1 2 119899) and 119906119895 is the unit vector of thecomplex Banach space 119883 Then 119865(119909) is an almost starlikemapping of complex order 120582 on the unit ball 119861 in complexBanach space119883

Theorem 22 Let 119891 be an almost starlike function of complexorder 120582 on119863 with 120582 isin C and Re 120582 le 0 Let

119865 (119909) = 119899minus1sum119895=1

(119891(1198791199091 (119909))1198791199091 (119909) )120573119895 119879119909119895 (119909) 119909119895 + 119909minus 119899minus1sum119895=1

119879119909119895 (119909) 119909119895 119909 isin 119861 (85)

where 1199091 isin 119861 1199091 = 1 120573119895 isin [0 1] (119891(119911)119911)120573119895 |119911=0 = 1 (119895 =1 2 119899 minus 1) and 1199091 119909119899 are linearly independent and119879119909119894 isin 119883lowast such that 119879119909119894 = 1 119879119909119894(119909119894) = 1 and 119879119909119894(119909119895) =0 (119894 = 119895) for forall119909119894 Then 119865(119909) is an almost starlike mapping ofcomplex order 120582 on the unit ball 119861 in complex Banach space119883

Proof From [24] we get 119865(119909) is a normalized biholomorphicmapping on 119861 and119909 119879119909 [119863119865 (119909)minus1 119865 (119909)]= 1199092

+ 119899minus1sum119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162 [ 119891 (1198791199091 (119909))1198791199091 (119909) 1198911015840 (1198791199091 (119909)) minus 1] (86)

Since119891 is an almost starlike function of complex order120582 thenR[[(1 minus 120582) 119891 (119879119909119895 (119909))119879119909119895 (119909) 1198911015840 (119879119909119895 (119909))]] ge minusR120582 (87)

Therefore

R (1 minus 120582) 119879119909 [119863119865 (119909)minus1 119865 (119909)] = (1 minusR120582) 119909+ 119899minus1sum119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162119909sdotR(1 minus 120582) [ 119891 (1198791199091 (119909))1198791199091 (119909) 1198911015840 (1198791199091 (119909)) minus 1]ge (1 minusR120582) 119909 + 119899minus1sum

119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162119909sdot [R (minus120582) minus (1 minusR120582)] = minusR120582 119909+ 1119909 [[1199092 minus

119899minus1sum119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162]] ge minusR120582 119909

(88)

which lead to the desired conclusion by Definition 5

Similar to Theorem 22 we can get the following conclu-sion

Theorem 23 Let 119891 be an almost starlike function of complexorder 120582 on119863 with 120582 isin C Re 120582 le 0 Let119865 (119909) = Φ120573 (119891) (119909)= 119891 (1198791199091 (119909)) 1199091

+ (119891 (1198791199091 (119909))1198791199091 (119909) )120573 (119909 minus 1198791199091 (119909) 1199091) (89)

where 0 le 120573 le 1 1199091 isin 119861 1199091 = 1 (119891(1199111)1199111)120573|1199111=0 = 1 and119891(1199111) = 0 for 1199111 = 0 Then 119865(119909) is an almost starlike mappingof complex order 120582 on 119861Remark 24 Setting 120582 = 120572(120572 minus 1) in Theorems 19ndash23 weget the corresponding results with respect to almost starlikemappings of order 120572

Discrete Dynamics in Nature and Society 11

Competing Interests

The authors declare that they have no conflict of interests

Acknowledgments

This work is supported by NSF of China (no U1204618) andScience and Technology Research Projects of Henan Provin-cial Education Department (nos 14B110015 and 14B110016)

References

[1] C Pommerenke Univalent Functions Vandenhoeck amp Ru-precht Gottingen Germany 1975

[2] H Cartan ldquoSur la possibilite drsquoentendre aux functions de plus-ieurs variables complexes la theorie des functions univalentsrdquoin Lecons sur les Functions Univalents onMutivalents P MontelEd Gauthier-Villar Paris France 1933

[3] R W Barnard C H FitzGerald and S Gong ldquoA distortionthoerem for biholomorphicmappings inC2rdquoTransactions of theAmerican Mathematical Society vol 344 pp 902ndash924 1994

[4] T S Liu andW J Zhang ldquoA distortion thoerem for biholomor-phic convex mappings in C119899rdquo Chinese Journal of ContemporaryMathematics vol 20 no 3 pp 421ndash430 1999

[5] S Gong S K Wang and Q H Yu ldquoBiholomorphic convexmappings of ball in C119899rdquo Pacific Journal of Mathematics vol 161no 2 pp 287ndash306 1993

[6] S Gong and T S Liu ldquoDistortion theorems for biholomorphicconvex mappings on bounded convex circular domainsrdquo Chi-nese Annals of Mathematics Series B vol 20 no 3 pp 297ndash3041999

[7] I Graham and D Varolin ldquoBloch constants in one and severalvariablesrdquoPacific Journal ofMathematics vol 174 no 2 pp 347ndash357 1996

[8] P L DurenUnivalent Functions Springer New York NY USA1983

[9] T H Gronwall ldquoSome remarks on conformal representationrdquoAnnals of Mathematics vol 16 no 1ndash4 pp 72ndash76 191415

[10] M I Robertson ldquoOn the theory of univalent functionsrdquo Annalsof Mathematics vol 37 no 2 pp 374ndash408 1936

[11] A V Boyd ldquoCoefficient estimates for starlike functions of order120572rdquo Proceedings of the AmericanMathematical Society vol 17 pp1016ndash1018 1966

[12] H Silverman ldquoUnivalent functions with negative coefficientsrdquoProceedings of the American Mathematical Society vol 51 pp109ndash116 1975

[13] H M Srivastava and S Owa ldquoCertain subclasses of starlikefunctions Irdquo Journal ofMathematical Analysis and Applicationsvol 161 no 2 pp 405ndash415 1991

[14] K A Roper and T J Suffridge ldquoConvex mappings on the unitball ofC119899rdquo Journal drsquoAnalyseMathematique vol 65 pp 333ndash3471995

[15] I Graham and G Kohr ldquoUnivalent mappings associated withthe Roper-Suffridge extension operatorrdquo Journal drsquoAnalyseMathematique vol 81 pp 331ndash342 2000

[16] S Gong and T S Liu ldquoThe generalized Roper-Suffridge exten-sion operatorrdquo Journal of Mathematical Analysis and Applica-tions vol 284 no 2 pp 425ndash434 2003

[17] G Kohr ldquoLoewner chains and a modification of the Roper-Suffridge extension operatorrdquo Mathematica vol 71 no 1 pp41ndash48 2006

[18] J R Muir ldquoA class of Loewner chain preserving extensionoperatorsrdquo Journal of Mathematical Analysis and Applicationsvol 337 no 2 pp 862ndash879 2008

[19] Y H Zhao Almost Starlike Mappings of Complex Order 120582 onthe Unit Ball of a Complex Banach Space Zhejiang NormalUniversity Jinhua China 2013

[20] L V Ahlfors Complex Analysis McGraw-Hill New York NYUSA 3rd edition 1978

[21] L Bieberbach ldquoUber einige extremal problem in gebiete derkonformen abbildungrdquo Mathematische Annalen vol 77 no 2pp 153ndash172 1916

[22] T Hayami S Owa and H M Srivastava ldquoCoefficient inequal-ities for certain classes of analytic and univalent functionsrdquoJournal of Inequalities in Pure and Applied Mathematics vol 8no 4 pp 1ndash10 2007

[23] M S Liu and Y C Zhu ldquoGeneralized Roper-Suffridge oper-ators on bounded and complete Reinhardt domainsrdquo Sciencein China vol 37 no 10 pp 1193ndash1206 2007

[24] Y Y Cui and C J Wang ldquoThe generalized Roper-Suffridgeextension operators in complex Banach spacerdquo Journal ofMathematics vol 34 no 3 pp 515ndash520 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 3

Let

119892 (119911) = 1199111198911015840 (119911)(1 minus 120582) 119891 (119911) 119911 isin 119863 0 11 minus 120582 119911 = 0 (12)

Therefore |2R(minus120582)119892(119911)minus1| le 1 Let119901(119911) = 2R(minus120582)119892(119911)minus1 We have |119901(119911)| le 1 and 119901(0) = 2R(minus120582)(1 minus 120582) minus 1 Letℎ (119911) = 119901 (119911) minus 119901 (0)1 minus 119901 (0)119901 (119911) 119911 isin 119863 (13)

Then ℎ(0) = 0 and |ℎ(119911)| lt 1 By Schwarz lemma we obtain|ℎ(119911)| le |119911| which follows10038161003816100381610038161003816100381610038161003816100381610038161003816 2R (minus120582) 119892 (119911) minus 2R (minus120582) (1 minus 120582)1 minus (2R (minus120582) (1 minus 120582) minus 1) [2R (minus120582) 119892 (119911) minus 1] 10038161003816100381610038161003816100381610038161003816100381610038161003816le |119911| (14)

For

1 minus (2R (minus120582)1 minus 120582 minus 1) [2R (minus120582) 119892 (119911) minus 1]= 1 + 1 + 1205821 minus 120582 [2R (minus120582) 119892 (119911) minus 1]= minus2R120582 [(1 + 120582) 119892 (119911) + 1]1 minus 120582

(15)

then by (14) we obtain10038161003816100381610038161003816100381610038161003816 (1 minus 120582) 119892 (119911) minus 1(1 + 120582) 119892 (119911) + 1 10038161003816100381610038161003816100381610038161003816 le |119911| (16)

which follows100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911) 119891 (119911) minus 11199111198911015840 (119911) 119891 (119911) + (1 minus 120582) (1 + 120582) 100381610038161003816100381610038161003816100381610038161003816 le 10038161003816100381610038161003816100381610038161003816 1 + 1205821 minus 12058210038161003816100381610038161003816100381610038161003816 |119911| (17)

By Lemma6we get that 1199111198911015840(119911)119891(119911) is in a circlewhose centeris 119886 and whose radius is 120588 where

119886 = 1 + |(1 + 120582) (1 minus 120582)|2 |119911|2 ((1 minus 120582) (1 + 120582))1 minus |(1 + 120582) (1 minus 120582)|2 |119911|2 120588 = |(1 + 120582) (1 minus 120582)| |119911| |1 + (1 minus 120582) (1 + 120582)|100381610038161003816100381610038161 minus |(1 + 120582) (1 minus 120582)|2 |119911|210038161003816100381610038161003816= 2 |119911| |1 minus 120582||1 minus 120582|2 minus |1 + 120582|2 |119911|2

(18)

So we obtain

R119886 minus 120588 le R1199111198911015840 (119911)119891 (119911) le R119886 + 120588 (19)

By direct computation we have

R119886 minus 120588 = 1 + |(1 + 120582) (1 minus 120582)|2 |119911|2R ((1 minus 120582) (1 + 120582))1 minus |(1 + 120582) (1 minus 120582)|2 |119911|2 minus 2 |119911| |1 minus 120582||1 minus 120582|2 minus |1 + 120582|2 |119911|2= 1 + |(1 + 120582) (1 minus 120582)|2 |119911|2 ((1 minus |120582|2) (1 + |120582|2 + 2R120582))1 minus |(1 + 120582) (1 minus 120582)|2 |119911|2 minus 2 |119911| |1 minus 120582||1 minus 120582|2 minus |1 + 120582|2 |119911|2= |1 + 120582|2 ((1 minus |120582|2) (1 + |120582|2 + 2R120582)) |119911|2 minus 2 |1 minus 120582| |119911| + |1 minus 120582|2|1 minus 120582|2 minus |1 + 120582|2 |119911|2

(20)

Similarly we can get

R119886 + 120588 = |1 + 120582|2 ((1 minus |120582|2) (1 + |120582|2 + 2R120582)) |119911|2 + 2 |1 minus 120582| |119911| + |1 minus 120582|2|1 minus 120582|2 minus |1 + 120582|2 |119911|2 = 119898 |119911|2 + 2 |1 minus 120582| |119911| + |1 minus 120582|2|1 minus 120582|2 minus |1 + 120582|2 |119911|2 (21)

where119898 = |1 + 120582|2((1 minus |120582|2)(1 + |120582|2 + 2R120582)) Let119891 (119909) = 1198981199092 + 2 |1 minus 120582| 119909 + |1 minus 120582|2|1 minus 120582|2 minus |1 + 120582|2 1199092 119909 isin [0 1) (22)

Immediately we have

1198911015840 (119909) = 119875(|1 minus 120582|2 minus |1 + 120582|2 1199092)2 (23)

4 Discrete Dynamics in Nature and Society

where119875 = (2119898119909 + 2 |1 minus 120582|) (|1 minus 120582|2 minus |1 + 120582|2 1199092) + (1198981199092+ 2 |1 minus 120582| 119909 + |1 minus 120582|2) 2 |1 + 120582|2 119909 = 2 |1 minus 120582| |1+ 120582|2 1199092 + 2 |1 minus 120582|2 (119898 + |1 + 120582|2) 119909 + 2 |1 minus 120582|3= 2 |1 minus 120582| [|1 + 120582|2 1199092 + |1 minus 120582| |1 + 120582|2sdot 2 (1 +R120582)1 + |120582|2 + 2R120582119909 + |1 minus 120582|2] = 2 |1 minus 120582| |1 + 120582|2sdot [[(119909 + |1 minus 120582| (1 +R120582)1 + |120582|2 + 2R120582 )2 + |1 minus 120582|2|1 + 120582|2minus |1 minus 120582|2 (1 +R120582)2(1 + |120582|2 + 2R120582)2]] = 2 |1 minus 120582| |1 + 120582|2(119909+ |1 minus 120582| (1 +R120582)1 + |120582|2 + 2R120582 )2+ |1 minus 120582|2|1 + 120582|2 (1 + |120582|2 + 2R120582)2 [(1 + |120582|2 + 2R120582)2minus |1 + 120582|2 (1 +R120582)2] = 2 |1 minus 120582| |1 + 120582|2 [[(119909+ |1 minus 120582| (1 +R120582)1 + |120582|2 + 2R120582 )2+ |1 minus 120582|2 (I120582)2|1 + 120582|2 (1 + |120582|2 + 2R120582)2]] ge 0

(24)

which follows 1198911015840(119909) ge 0 Therefore 119891(119909) is monotoneincreasing So

R119886 + 120588 = 119898 |119911|2 + 2 |1 minus 120582| |119911| + |1 minus 120582|2|1 minus 120582|2 minus |1 + 120582|2 |119911|2le 1198981199032 + 2 |1 minus 120582| 119903 + |1 minus 120582|2|1 minus 120582|2 minus |1 + 120582|2 1199032 (25)

where 119911 = 119903119890119894120579 120579 isin [0 2120587) By (19) we haveR(1199111198911015840(119911)119891(119911)) =119903(120597120597119903) ln |119891(119911)| le R119886 + 120588 Therefore120597120597119903 ln 1003816100381610038161003816119891 (119911)1003816100381610038161003816 le 1119903 1198981199032 + 2 |1 minus 120582| 119903 + |1 minus 120582|2|1 minus 120582|2 minus |1 + 120582|2 1199032= 1119903 (|1 minus 120582|2 minus |1 + 120582|2 1199032) [ |120582|2 minus 11 + |120582|2 + 2R120582 (|1minus 120582|2 minus |1 + 120582|2 1199032) + 1 minus |120582|21 + |120582|2 + 2R120582 |1 minus 120582|2 + |1

minus 120582|2 + 2 |1 minus 120582| 119903] = 1119903 [ |120582|2 minus 11 + |120582|2 + 2R120582 + 2 |1minus 120582| 119903 + (1 +R120582) |1 minus 120582| (1 + |120582|2 + 2R120582)|1 minus 120582|2 minus |1 + 120582|2 1199032 ]

(26)

Integrating both ends of (26) on [120576 119903] we obtainint119903120576

120597120597119905 ln 10038161003816100381610038161003816119891 (119905119890119894120579)10038161003816100381610038161003816 119889119905 le |120582|2 minus 11 + |120582|2 + 2R120582 int119903120576

119889119905119905 + 2 |1minus 120582| [int119903

120576

119889119905|1 minus 120582|2 minus |1 + 120582|2 1199052+ (1 +R120582) |1 minus 120582|1 + |120582|2 + 2R120582 int119903

120576

1119905 119889119905|1 minus 120582|2 minus |1 + 120582|2 1199052]= |120582|2 minus 11 + |120582|2 + 2R120582 ln 119905|119903120576 + 2 |1 minus 120582| 12 10038161003816100381610038161 minus 12058221003816100381610038161003816sdot ln 10038161003816100381610038161003816100381610038161003816 119905 + |(1 minus 120582) (1 + 120582)|119905 minus |(1 minus 120582) (1 + 120582)| 1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119903120576 + 2 (1 +R120582) |1 minus 120582|21 + |120582|2 + 2R120582sdot 12 |1 minus 120582|2 ln 119905210038161003816100381610038161003816|1 minus 120582|2 minus |1 + 120582|2 119905210038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816100381610038161003816119903

120576

(27)

Therefore

ln 1003816100381610038161003816119891 (119911)1003816100381610038161003816 minus ln 10038161003816100381610038161003816119891 (120576119890119894120579)10038161003816100381610038161003816 le |120582|2 minus 11 + |120582|2 + 2R120582 (ln 119903minus ln 120576) + 1|1 + 120582| (ln 10038161003816100381610038161003816100381610038161003816 119903 + |(1 minus 120582) (1 + 120582)|119903 minus |(1 minus 120582) (1 + 120582)| 10038161003816100381610038161003816100381610038161003816minus ln

10038161003816100381610038161003816100381610038161003816 120576 + |(1 minus 120582) (1 + 120582)|120576 minus |(1 minus 120582) (1 + 120582)| 10038161003816100381610038161003816100381610038161003816)+ (1 +R120582)1 + |120582|2 + 2R120582 (ln 119903210038161003816100381610038161003816|1 minus 120582|2 minus |1 + 120582|2 119903210038161003816100381610038161003816minus ln 120576210038161003816100381610038161003816|1 minus 120582|2 minus |1 + 120582|2 120576210038161003816100381610038161003816)

(28)

Since

ln 10038161003816100381610038161003816119891 (120576119890119894120579)10038161003816100381610038161003816 minus |120582|2 minus 11 + |120582|2 + 2R120582 ln 120576minus 1 +R1205821 + |120582|2 + 2R120582 ln 1205762 = ln

10038161003816100381610038161003816119891 (120576119890119894120579)10038161003816100381610038161003816120576 997888rarr 0(120576 997888rarr 0)

(29)

Discrete Dynamics in Nature and Society 5

setting 120576 rarr 0 in (28) we have

ln 1003816100381610038161003816119891 (119911)1003816100381610038161003816 le |120582|2 minus 11 + |120582|2 + 2R120582 ln 119903 + 1|1 + 120582|sdot ln 10038161003816100381610038161003816100381610038161003816 119903 + |(1 minus 120582) (1 + 120582)|119903 minus |(1 minus 120582) (1 + 120582)| 10038161003816100381610038161003816100381610038161003816+ 1 +R1205821 + |120582|2 + 2R120582 (ln 119903210038161003816100381610038161003816|1 minus 120582|2 minus |1 + 120582|2 119903210038161003816100381610038161003816+ ln |1 minus 120582|2)

(30)

Therefore1003816100381610038161003816119891 (119911)1003816100381610038161003816 le 119903(|120582|2minus1)(1+|120582|2+2R120582)sdot 10038161003816100381610038161003816100381610038161003816 119903 + |(1 minus 120582) (1 + 120582)|119903 minus |(1 minus 120582) (1 + 120582)| 100381610038161003816100381610038161003816100381610038161|1+120582|sdot ( |1 minus 120582|2 119903210038161003816100381610038161003816|1 minus 120582|2 minus |1 + 120582|2 119903210038161003816100381610038161003816)

(1+R120582)(1+|120582|2+2R120582)

= 119903 10038161003816100381610038161003816100381610038161003816 119903 + |(1 minus 120582) (1 + 120582)|119903 minus |(1 minus 120582) (1 + 120582)| 100381610038161003816100381610038161003816100381610038161|1+120582|sdot ( 1100381610038161003816100381610038161 minus |(1 + 120582) (1 minus 120582)|2 119903210038161003816100381610038161003816)

(1+R120582)(1+|120582|2+2R120582)

= 119903 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)(1+|120582|2+2R120582)sdot 10038161003816100381610038161003816100381610038161003816119903 minus 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 1205821003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816minus1|1+120582|minus(1+R120582)(1+|120582|2+2R120582)sdot 10038161003816100381610038161003816100381610038161003816119903 + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161|1+120582|minus(1+R120582)(1+|120582|2+2R120582)= 119903 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2sdot 10038161003816100381610038161003816100381610038161003816119903 minus 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 1205821003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(minus|1+120582|minus1minusR120582)|1+120582|2sdot 10038161003816100381610038161003816100381610038161003816119903 + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 1205821003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(|1+120582|minus1minusR120582)|1+120582|2 = |119911|sdot 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2sdot 10038161003816100381610038161003816100381610038161003816|119911| minus 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 1205821003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(minus|1+120582|minus1minusR120582)|1+120582|2sdot (|119911| + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816)(|1+120582|minus1minusR120582)|1+120582|2

(31)

Obviously |(1 minus 120582)(1 + 120582)| ge 1 forR120582 le 0 Thus |119911| le |(1 minus120582)(1 + 120582)| So we obtain1003816100381610038161003816119891 (119911)1003816100381610038161003816 le |119911| 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2

sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 minus |119911|)(minus|1+120582|minus1minusR120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 + |119911|)(|1+120582|minus1minusR120582)|1+120582|2

(32)

Similarly by (19) we can get

1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge |119911| 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 minus |119911|)(|1+120582|minus1minusR120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 + |119911|)(minus|1+120582|minus1minusR120582)|1+120582|2

(33)

Theorem8 Let119891(119911) be an almost starlike function of complexorder 120582 on119863 with 120582 = minus1 Then

1 minus |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le 1 + |119911| |119911| 119890minus|119911| le 1003816100381610038161003816119891 (119911)1003816100381610038161003816 le |119911| 119890|119911|(1 minus |119911|) 119890minus|119911| le 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 le (1 + |119911|) 119890|119911|

(34)

Proof By (14) we have |2119892(119911) minus 1| le |119911| which follows|1199111198911015840(119911)119891(119911) minus 1| le |119911| Therefore

1 minus |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le 1 + |119911| 1 minus |119911| le R

1199111198911015840 (119911)119891 (119911) le 1 + |119911| (35)

For |119911| = 119903 we have1 minus 119903 le 119903 120597120597119903 ln 1003816100381610038161003816119891 (119911)1003816100381610038161003816 = R

1199111198911015840 (119911)119891 (119911) le 1 + 119903 (36)

Similar to Theorem 7 we get

int119903120576

119889119905119905 minus int119903120576119889119905 le int119903

120576

120597120597119905 ln 10038161003816100381610038161003816119891 (119905119890119894120579)10038161003816100381610038161003816 119889119905le int119903120576

119889119905119905 + int119903120576119889119905 (37)

6 Discrete Dynamics in Nature and Society

Let 120576 rarr 0 we obtain |119911|119890minus|119911| le |119891(119911)| le |119911|119890|119911| Hence (1 minus|119911|)119890minus|119911| le |1198911015840(119911)| le (1 + |119911|)119890|119911|Setting 120582 = 120572(120572 minus 1) in Theorems 7 and 8 we get the

following corollary

Corollary 9 Let 119891(119911) be an almost starlike function of order120572 Then |119911|(1 + (1 minus 2120572) |119911|)2(1minus120572)(1minus2120572) le 1003816100381610038161003816119891 (119911)1003816100381610038161003816le |119911|(1 minus (1 minus 2120572) |119911|)2(1minus120572)(1minus2120572) 120572 isin [0 1) 12

|119911| 119890minus|119911| le 1003816100381610038161003816119891 (119911)1003816100381610038161003816 le |119911| 119890|119911| 120572 = 12 1 minus |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le 1 + |119911| (1 minus |119911|) 119890minus|119911| le 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 le (1 + |119911|) 119890|119911|

120572 = 12

(38)

Theorem 10 Let 119891(119911) be an almost starlike function ofcomplex order 120582 on119863 with 120582 isin C minus1 Re 120582 le 0 Then1 minus |119911|1 + |(1 + 120582) (1 minus 120582)| |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816le 1 + |119911|1 minus |(1 + 120582) (1 minus 120582)| |119911|

(1 minus |119911|) 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2+1sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 minus |119911|)(|1+120582|minus1minusR120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 + |119911|)(minus|1+120582|minus1minusR120582)|1+120582|2minus1 le 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816le (1 + |119911|) 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2+1sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 + |119911|)(|1+120582|minus1minusR120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 minus |119911|)(minus|1+120582|minus1minusR120582)|1+120582|2minus1

(39)

Proof By (17) we have100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) minus 1100381610038161003816100381610038161003816100381610038161003816 le 10038161003816100381610038161003816100381610038161003816 1 + 1205821 minus 12058210038161003816100381610038161003816100381610038161003816 |119911| 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) + 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038161003816 (40)

Since 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 minus 1 le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) minus 1100381610038161003816100381610038161003816100381610038161003816 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) + 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038161003816 le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 (41)

then we have100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 minus 1 le (100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816) 10038161003816100381610038161003816100381610038161003816 1 + 1205821 minus 12058210038161003816100381610038161003816100381610038161003816 |119911| (42)

which follows100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le 1 + |119911|1 minus |(1 + 120582) (1 minus 120582)| |119911| (43)

On the other hand we also have

1 minus 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le (100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816) 10038161003816100381610038161003816100381610038161003816 1 + 1205821 minus 12058210038161003816100381610038161003816100381610038161003816 |119911| (44)

which leads to100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 ge 1 minus |119911|1 + |(1 + 120582) (1 minus 120582)| |119911| (45)

Hence 1 minus |119911|1 + |(1 + 120582) (1 minus 120582)| |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816le 1 + |119911|1 minus |(1 + 120582) (1 minus 120582)| |119911| (46)

Then by Theorem 7 we obtain the desired conclusion (39)

Setting 120582 = 120572(120572 minus 1) inTheorem 10 we get the followingresult

Corollary 11 Let 119891(119911) be an almost starlike function of order120572 with 120572 isin [0 1) 12 Then1 minus |119911|1 + |1 minus 2120572| |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le 1 + |119911|1 minus |1 minus 2120572| |119911| 1 minus |119911|1 + |1 minus 2120572| |119911| |119911|(1 + (1 minus 2120572) |119911|)2(1minus120572)(1minus2120572)le 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816le 1 + |119911|1 minus |1 minus 2120572| |119911| |119911|(1 minus (1 minus 2120572) |119911|)2(1minus120572)(1minus2120572) (47)

3 Coefficient Estimates of Almost StarlikeFunctions of Complex Order 120582

Lemma 12 (see [21]) Let 119901(119911) = 1 + suminfin119899=1 119888119899119911119899 be aholomorphic function on 119863 with R119901(119911) gt 0 Then |119888119899| le2 (119899 = 1 2 )

Discrete Dynamics in Nature and Society 7

Lemma 13 (see [22]) Let119901(119911) = 1+suminfin119899=1 119888119899119911119899 be holomorphicon119863 ThenR119901(119911) gt 0 if and only if

119901 (119911) = 120585 minus 1120585 + 1 120585 isin 120597119863 (48)

Theorem 14 Let 119891(119911) = 119911 + suminfin119899=2 119886119899119911119899 be an almost starlikefunction of complex order 120582 with 120582 isin C Re 120582 le 0 on119863 Then10038161003816100381610038161198861198991003816100381610038161003816 le 119899minus1prod

119895=2

( 2|1 minus 120582| + 119895 minus 2119895 minus 1) 119899 = 2 3 (49)

Proof Since 119891(119911) is an almost starlike function of complexorder 120582 then

R [(1 minus 120582) 119891 (119911)1199111198911015840 (119911) + 120582] gt 0 119911 isin 119863 (50)

Let 119901(119911) = (1 minus 120582)(119891(119911)1199111198911015840(119911)) + 120582 Then we have1199111198911015840 (119911) (119901 (119911) minus 120582) = (1 minus 120582) 119891 (119911) (51)

and 119901(0) = 1 andR119901(119911) gt 0 Let 119901(119911) = 1 + suminfin119899=1 119888119899119911119899 Then10038161003816100381610038161198881198991003816100381610038161003816 le 2 (119899 = 1 2 ) (52)

by Lemma 12 For 119891(119911) = 119911 + suminfin119899=2 119886119899119911119899 (51) follows119911(1 + infinsum

119899=2

119899119886119899119911119899minus1)(1 minus 120582 + infinsum119899=1

119888119899119911119899)= (1 minus 120582)(119911 + infinsum

119899=2

119886119899119911119899) (53)

Therefore(1 minus 120582) 119911 + [1198881 + 21198862 (1 minus 120582)] 1199112 + sdot sdot sdot + [119888119899minus1+ 21198862119888119899minus2 + sdot sdot sdot + (119899 minus 1) 119886119899minus11198881 + 119899119886119899 (1 minus 120582)] 119911119899+ sdot sdot sdot = (1 minus 120582) 119911 + 1198862 (1 minus 120582) 1199112 + sdot sdot sdot + 119886119899 (1 minus 120582)sdot 119911119899 + sdot sdot sdot (54)

So 119886119899 = 119888119899minus1 + 21198862119888119899minus2 + sdot sdot sdot + (119899 minus 1) 119886119899minus11198881(119899 minus 1) (120582 minus 1) (55)

By (52) we get10038161003816100381610038161198861198991003816100381610038161003816 le 2 [1 + 2 100381610038161003816100381611988621003816100381610038161003816 + sdot sdot sdot + (119899 minus 1) 1003816100381610038161003816119886119899minus11003816100381610038161003816](119899 minus 1) |1 minus 120582| (56)

Bymathematical induction we obtain the desired conclusion

Theorem 15 Let 119891(119911) be an almost starlike function ofcomplex order 120582 with 120582 isin C Re 120582 le 0 on 119863 and let 119911 = 0be the zero of order 119896 + 1 (119896 isin N) of 119891(119911) minus 119911 Let 119891(119911) =119911 + suminfin119899=119896+1 119886119899119911119899 Then1003816100381610038161003816119886119899+11003816100381610038161003816 le 2|120582 + 119899| (119899 = 119896 119896 + 1 ) (57)

Proof Since 119891(119911) = 119911 + suminfin119899=119896+1 119886119899119911119899 similar to Theorem 14by (51) we obtain

119911(1 + infinsum119899=119896+1

119899119886119899119911119899minus1)(1 minus 120582 + infinsum119899=1

119888119899119911119899)= (1 minus 120582)(119911 + infinsum

119899=119896+1

119886119899119911119899) (58)

which follows(1 minus 120582) 119911 + 11988811199112 + sdot sdot sdot + 119888119896minus1119911119896+ [119888119896 + (119896 + 1) 119886119896+1] 119911119896+1 + sdot sdot sdot+ [119888119899 + (119899 + 1) 119886119899+1] 119911119899+1 + sdot sdot sdot= (1 minus 120582) 119911 + (1 minus 120582) 119886119896+1119911119896+1 + sdot sdot sdot+ (1 minus 120582) 119886119899+1119911119899+1 + sdot sdot sdot (59)

Therefore 1198881 = sdot sdot sdot = 119888119896minus1 = 0119888119899 = (minus120582 minus 119899) 119886119899+1 (119899 = 119896 119896 + 1 ) (60)

By (52) we get the desired conclusion

Theorem 16 119891(119911) = 119911 + suminfin119899=2 119886119899119911119899 is an almost starlikefunction of complex order 120582 with 120582 isin C and Re 120582 le 0 on 119863if and only ifinfinsum119899=2

[(1 + (119899 minus 1) 120582) (120585 + 1) + 119899 (1 minus 120585)] 119886119899119911119899 = minus2(120585 isin 120597119863 119899 = 2 3 ) (61)

Proof Let 119901(119911) = (1 minus 120582)(119891(119911)1199111198911015840(119911)) + 120582 Since 119891(119911) isan almost starlike function of complex order 120582 then 119901(119911)is holomorphic on 119863 and 119901(0) = 0 and R119901(119911) gt 0 ByLemma 13 we get(1 minus 120582) 119891 (119911)1199111198911015840 (119911) + 120582 = 120585 minus 1120585 + 1 120585 isin 120597119863 (62)

which follows(1 minus 120582) (120585 + 1) 119891 (119911) + 120582 (120585 + 1) 1199111198911015840 (119911)= (120585 minus 1) 1199111198911015840 (119911) (63)

For 119891(119911) = 119911 + suminfin119899=2 119886119899119911119899 we have(1 minus 120582) (120585 + 1) (119911 + infinsum119899=2

119886119899119911119899)+ 120582 (120585 + 1) (119911 + infinsum

119899=2

119899119886119899119911119899)= (120585 minus 1) (119911 + infinsum

119899=2

119899119886119899119911119899) (64)

which leads to the desired conclusion

8 Discrete Dynamics in Nature and Society

Remark 17 Setting 120582 = 120572(120572 minus 1) in Theorems 14ndash16 weget the corresponding results for almost starlike functions oforder 1205724 The Invariance of Almost Starlike Mappings

of Complex Order 120582In this section we mainly discuss the invariance of almoststarlike mappings of complex order 120582 on Reinhardt domainsand on the unit ball in complex Banach spaces under somegeneralized Roper-Suffridge operators

Lemma 18 (see [23]) Let Ω sub C119899 be a bounded andcompletely Reinhardt domain whose Minkowski functional120588(119911) is 1198621 except for a lower-dimensional manifold Ω0 Thenfor forall119911 = (1199111 119911119899) isin Ω Ω0 we have120597120588 (119911)120597119911 = (120597120588 (119911)1205971199111 120597120588 (119911)1205971199112 120597120588 (119911)120597119911119899 )

120597120588 (119911)120597119911119895 119911119895 ge 0120588 (119911) = 2 119899sum

119895=1

120597120588 (119911)120597119911119895 119911119895(65)

Theorem 19 Let 119891(119911) be an almost starlike function of com-plex order 120582 on 119863 with 120582 isin C and Re 120582 le 0 Let Ω sub C119899 be abounded and completely Reinhardt domain whose Minkowskifunctional 120588(119911) is 1198621 on Ω 0 Let

119865 (119911) = (119903119891(1199111119903 ) (119903119891 (1199111119903)1199111 )1205732sdot 1199112 (119903119891 (1199111119903)1199111 )120573119899 119911119899)1015840

(66)

where 120573119895 isin [0 1] 119903 = sup|1199111| 119911 = (1199111 119911119899)1015840 isin Ω and(119903119891(1199111119903)1199111)120573119895 |1199111=0 = 1 (119895 = 2 119899) Then 119865(119911) is an almoststarlike mapping of complex order 120582 on ΩProof For

119865 (119911) = (119903119891(1199111119903 ) (119903119891 (1199111119903)1199111 )1205732sdot 1199112 (119903119891 (1199111119903)1199111 )120573119899 119911119899)1015840

(67)

immediately we have

119869119865 (119911) =(((((((((((

1198911015840 (1199111119903 ) 0 sdot sdot sdot 01205732119903 [1198911015840 (1199111119903)119891 (1199111119903) minus 1199031199111 ](119903119891 (1199111119903)1199111 )1205732 1199112 (119903119891 (1199111119903)1199111 )1205732 sdot sdot sdot 0

sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot120573119899119903 [1198911015840 (1199111119903)119891 (1199111119903) minus 1199031199111](119903119891 (1199111119903)1199111 )120573119899 119911119899 0 sdot sdot sdot (119903119891 (1199111119903)1199111 )120573119899

)))))))))))

(68)

Therefore

119869minus1119865 (119911) =(((((((((((

11198911015840 (1199111119903) 0 sdot sdot sdot 01205732119903 [ 11990311991111198911015840 (1199111119903) minus 1119891 (1199111119903)] 1199112 (119903119891 (1199111119903)1199111 )minus1205732 sdot sdot sdot 0

sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot120573119899119903 [ 11990311991111198911015840 (1199111119903) minus 1119891 (1199111119903)] 119911119899 0 sdot sdot sdot (119903119891 (1199111119903)1199111 )minus120573119899

)))))))))))

(69)

Discrete Dynamics in Nature and Society 9

Then 119869minus1119865 (119911) 119865 (119911)= (119903119891 (1199111119903)1198911015840 (1199111119903) (1 minus 1205732 + 1205732 119903119891 (1199111119903)11991111198911015840 (1199111119903))sdot 1199112 (1 minus 120573119899 + 120573119899 119903119891 (1199111119903)11991111198911015840 (1199111119903)) 119911119899)1015840

(70)

By Lemma 18 we get(1 minus 120582) 2120597120588120597119911 (119911) 119869minus1119865 (119911) 119865 (119911) = 2 (1 minus 120582)sdot [[ 119903119891 (1199111119903)11991111198911015840 (1199111119903) 120597120588 (119911)1205971199111 1199111+ 119899sum119895=2

(1 minus 120573119895 + 120573119895 119903119891 (1199111119903)11991111198911015840 (1199111119903)) 120597120588 (119911)120597119911119895 119911119895]] (71)

Since119891(119911) is an almost starlike function of complex order120582 thenR[(1 minus 120582) 119903119891 (1199111119903)11991111198911015840 (1199111119903)] ge minusR120582 (72)

Therefore by Lemma 18 we obtain

R [(1 minus 120582) 2120597120588120597119911 (119911) 119869minus1119865 (119911) 119865 (119911)] = R[(1 minus 120582)sdot 119903119891 (1199111119903)11991111198911015840 (1199111119903)] 2120597120588 (119911)1205971199111 1199111 + 119899sum

119895=2

(1 minus 120573119895)R (1 minus 120582)+ 120573119895R[(1 minus 120582) 119903119891 (1199111119903)11991111198911015840 (1199111119903)] 2120597120588 (119911)120597119911119895 119911119895 ge minusR120582sdot 2120597120588 (119911)1205971199111 1199111 + 119899sum

119895=2

(1 minus 120573119895) (1 minusR120582) + 120573119895 (minusR120582)sdot 2120597120588 (119911)120597119911119895 119911119895 = (minusR120582) [2120597120588 (119911)1205971199111 1199111 + 2120597120588 (119911)120597119911119895 119911119895]+ 119899sum119895=2

(1 minus 120573119895) 2120597120588 (119911)120597119911119895 119911119895 ge (minusR120582) 120588 (119911)

(73)

Hence 119865(119911) is an almost starlike mapping of complex order 120582onΩ by Definition 4

Theorem 20 Let 119891119895 (119895 = 1 2 119899) be almost starlikefunctions of complex order 120582 on 119863 with 120582 isin C and Re 120582 lt 0Let Ω sub C119899 be a bounded starlike and circular domain whoseMinkowski functional 120588(119911) is1198621 onΩ0 Let 119903119895 be the radiusof the disk 119880119895 = 119911119895 isin C 119911 = (1199111 119911119895 119911119899)1015840 isin Ω whosecenter is zero Let

119865 (119911) = 119911 119899prod119895=1

(119903119895119891119895 (119911119895119903119895)119911119895 )120582119894 (74)

where 120582119894119895 ge 0 sum119899119895=1 120582119894119895 = 1 and (119891119895(119911119895)119911119895)120582119894119895 |119911119895=0 = 1 (119894 119895 =1 2 119899)Then119865(119911) is an almost starlikemapping of complexorder 120582 on ΩProof Let

120577119895 = 119911119895119903119895 119892 (119911) = 119899prod

119895=1

(119891119895 (120577119895)120577119895 )120582119894 (75)

which follows 119865(119911) = 119911119892(119911) Then 119869119865(119911) = 119869119892(119911)119911 + 119892(119911) Bydirect computation we get

119869119892 (119911) 119911 = 119892 (119911) [[119899sum119895=1

1205821198951198911015840119895 (119911119895119903119895) 119911119895119903119895119891119895 (119911119895119903119895) minus 1]]= 119892 (119911) [[

119899sum119895=1

1205821198951198911015840119895 (120577119895) 120577119895119891119895 (120577119895) minus 1]] (76)

Since 119891119895 are almost starlike functions of complex order 120582 on119863 then

R[[(1 minus 120582) 119891119895 (120577119895)1205771198951198911015840119895 (120577119895)]] ge minusR120582 gt 0 (77)

Therefore

R[ 11 minus 120582 (1 + 119869119892 (119911) 119911119892 (119911) )]= R[[ 11 minus 120582 119899sum

119895=1

1205821198951198911015840119895 (120577119895) 120577119895119891119895 (120577119895) ]]= 119899sum119895=1

120582119895R[[ 11 minus 120582 1198911015840119895 (120577119895) 120577119895119891119895 (120577119895) ]] gt 0(78)

Then 119869119892(119911)119911+119892(119911) = 0 It is obvious that119865(0) = 0 and 119869119865(0) =119868 so 119865(119911) is a normalized locally biholomorphic mapping onΩ and

119869minus1119865 (119911) 119865 (119911) = 119892 (119911) 119911119869119892 (119911) 119911 + 119892 (119911) (79)

Let

119902119895 (120577119895) = 2Re (minus120582) 1198911015840119895 (120577119895) 120577119895(1 minus 120582) 119891119895 (120577119895) minus 1 (80)

(77) follows 100381610038161003816100381610038161003816100381610038161003816100381610038162Re (minus120582) 1198911015840119895 (120577119895) 120577119895(1 minus 120582) 119891119895 (120577119895) minus 110038161003816100381610038161003816100381610038161003816100381610038161003816 le 1 (81)

10 Discrete Dynamics in Nature and Society

that is |119902119895(120577119895)| le 1 For Re 120582 lt 0 by (76) (79) and Lemma 18we obtain1003816100381610038161003816100381610038161003816100381610038162R (minus120582) 1(1 minus 120582) (2120588 (119911)) (120597120588120597119911) (119911) 119869minus1119865 (119911) 119865 (119911)

minus 1100381610038161003816100381610038161003816100381610038161003816 =100381610038161003816100381610038161003816100381610038161003816100381610038162R (minus120582) 11 minus 120582 119899sum

119895=1

1205821198951198911015840119895 (120577119895) 120577119895119891119895 (120577119895) minus 110038161003816100381610038161003816100381610038161003816100381610038161003816= 10038161003816100381610038161003816100381610038161003816100381610038161003816 119899sum119895=1120582119895119902119895 (120577119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816 le 119899sum119895=1120582119895 = 1(82)

which leads to

R [(1 minus 120582) 2120597120588120597119911 (119911) 119869minus1119865 (119911) 119865 (119911)] ge minusR120582120588 (119911) (83)

Hence 119865(119911) is an almost starlike mapping of complex order 120582onΩ by Definition 4

Similar to Theorem 20 we can get the following conclu-sion

Theorem 21 Let 119891119895 (119895 = 1 2 119899) be almost starlikefunctions of complex order 120582 on 119863 with 120582 isin C and Re 120582 lt 0Let

119865 (119909) = 119909 119899prod119895=1

(119891119895 (119879119906119895 (119909))119879119906119895 (119909) )120582119895 119909 isin 119861 (84)

where 120582119894119895 ge 0 sum119899119895=1 120582119894119895 = 1 (119891119895(119909119895)119909119895)120582119894119895 |119909119895=0 = 1 (119909119895 =119879119906119895(119909) 119894 119895 = 1 2 119899) and 119906119895 is the unit vector of thecomplex Banach space 119883 Then 119865(119909) is an almost starlikemapping of complex order 120582 on the unit ball 119861 in complexBanach space119883

Theorem 22 Let 119891 be an almost starlike function of complexorder 120582 on119863 with 120582 isin C and Re 120582 le 0 Let

119865 (119909) = 119899minus1sum119895=1

(119891(1198791199091 (119909))1198791199091 (119909) )120573119895 119879119909119895 (119909) 119909119895 + 119909minus 119899minus1sum119895=1

119879119909119895 (119909) 119909119895 119909 isin 119861 (85)

where 1199091 isin 119861 1199091 = 1 120573119895 isin [0 1] (119891(119911)119911)120573119895 |119911=0 = 1 (119895 =1 2 119899 minus 1) and 1199091 119909119899 are linearly independent and119879119909119894 isin 119883lowast such that 119879119909119894 = 1 119879119909119894(119909119894) = 1 and 119879119909119894(119909119895) =0 (119894 = 119895) for forall119909119894 Then 119865(119909) is an almost starlike mapping ofcomplex order 120582 on the unit ball 119861 in complex Banach space119883

Proof From [24] we get 119865(119909) is a normalized biholomorphicmapping on 119861 and119909 119879119909 [119863119865 (119909)minus1 119865 (119909)]= 1199092

+ 119899minus1sum119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162 [ 119891 (1198791199091 (119909))1198791199091 (119909) 1198911015840 (1198791199091 (119909)) minus 1] (86)

Since119891 is an almost starlike function of complex order120582 thenR[[(1 minus 120582) 119891 (119879119909119895 (119909))119879119909119895 (119909) 1198911015840 (119879119909119895 (119909))]] ge minusR120582 (87)

Therefore

R (1 minus 120582) 119879119909 [119863119865 (119909)minus1 119865 (119909)] = (1 minusR120582) 119909+ 119899minus1sum119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162119909sdotR(1 minus 120582) [ 119891 (1198791199091 (119909))1198791199091 (119909) 1198911015840 (1198791199091 (119909)) minus 1]ge (1 minusR120582) 119909 + 119899minus1sum

119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162119909sdot [R (minus120582) minus (1 minusR120582)] = minusR120582 119909+ 1119909 [[1199092 minus

119899minus1sum119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162]] ge minusR120582 119909

(88)

which lead to the desired conclusion by Definition 5

Similar to Theorem 22 we can get the following conclu-sion

Theorem 23 Let 119891 be an almost starlike function of complexorder 120582 on119863 with 120582 isin C Re 120582 le 0 Let119865 (119909) = Φ120573 (119891) (119909)= 119891 (1198791199091 (119909)) 1199091

+ (119891 (1198791199091 (119909))1198791199091 (119909) )120573 (119909 minus 1198791199091 (119909) 1199091) (89)

where 0 le 120573 le 1 1199091 isin 119861 1199091 = 1 (119891(1199111)1199111)120573|1199111=0 = 1 and119891(1199111) = 0 for 1199111 = 0 Then 119865(119909) is an almost starlike mappingof complex order 120582 on 119861Remark 24 Setting 120582 = 120572(120572 minus 1) in Theorems 19ndash23 weget the corresponding results with respect to almost starlikemappings of order 120572

Discrete Dynamics in Nature and Society 11

Competing Interests

The authors declare that they have no conflict of interests

Acknowledgments

This work is supported by NSF of China (no U1204618) andScience and Technology Research Projects of Henan Provin-cial Education Department (nos 14B110015 and 14B110016)

References

[1] C Pommerenke Univalent Functions Vandenhoeck amp Ru-precht Gottingen Germany 1975

[2] H Cartan ldquoSur la possibilite drsquoentendre aux functions de plus-ieurs variables complexes la theorie des functions univalentsrdquoin Lecons sur les Functions Univalents onMutivalents P MontelEd Gauthier-Villar Paris France 1933

[3] R W Barnard C H FitzGerald and S Gong ldquoA distortionthoerem for biholomorphicmappings inC2rdquoTransactions of theAmerican Mathematical Society vol 344 pp 902ndash924 1994

[4] T S Liu andW J Zhang ldquoA distortion thoerem for biholomor-phic convex mappings in C119899rdquo Chinese Journal of ContemporaryMathematics vol 20 no 3 pp 421ndash430 1999

[5] S Gong S K Wang and Q H Yu ldquoBiholomorphic convexmappings of ball in C119899rdquo Pacific Journal of Mathematics vol 161no 2 pp 287ndash306 1993

[6] S Gong and T S Liu ldquoDistortion theorems for biholomorphicconvex mappings on bounded convex circular domainsrdquo Chi-nese Annals of Mathematics Series B vol 20 no 3 pp 297ndash3041999

[7] I Graham and D Varolin ldquoBloch constants in one and severalvariablesrdquoPacific Journal ofMathematics vol 174 no 2 pp 347ndash357 1996

[8] P L DurenUnivalent Functions Springer New York NY USA1983

[9] T H Gronwall ldquoSome remarks on conformal representationrdquoAnnals of Mathematics vol 16 no 1ndash4 pp 72ndash76 191415

[10] M I Robertson ldquoOn the theory of univalent functionsrdquo Annalsof Mathematics vol 37 no 2 pp 374ndash408 1936

[11] A V Boyd ldquoCoefficient estimates for starlike functions of order120572rdquo Proceedings of the AmericanMathematical Society vol 17 pp1016ndash1018 1966

[12] H Silverman ldquoUnivalent functions with negative coefficientsrdquoProceedings of the American Mathematical Society vol 51 pp109ndash116 1975

[13] H M Srivastava and S Owa ldquoCertain subclasses of starlikefunctions Irdquo Journal ofMathematical Analysis and Applicationsvol 161 no 2 pp 405ndash415 1991

[14] K A Roper and T J Suffridge ldquoConvex mappings on the unitball ofC119899rdquo Journal drsquoAnalyseMathematique vol 65 pp 333ndash3471995

[15] I Graham and G Kohr ldquoUnivalent mappings associated withthe Roper-Suffridge extension operatorrdquo Journal drsquoAnalyseMathematique vol 81 pp 331ndash342 2000

[16] S Gong and T S Liu ldquoThe generalized Roper-Suffridge exten-sion operatorrdquo Journal of Mathematical Analysis and Applica-tions vol 284 no 2 pp 425ndash434 2003

[17] G Kohr ldquoLoewner chains and a modification of the Roper-Suffridge extension operatorrdquo Mathematica vol 71 no 1 pp41ndash48 2006

[18] J R Muir ldquoA class of Loewner chain preserving extensionoperatorsrdquo Journal of Mathematical Analysis and Applicationsvol 337 no 2 pp 862ndash879 2008

[19] Y H Zhao Almost Starlike Mappings of Complex Order 120582 onthe Unit Ball of a Complex Banach Space Zhejiang NormalUniversity Jinhua China 2013

[20] L V Ahlfors Complex Analysis McGraw-Hill New York NYUSA 3rd edition 1978

[21] L Bieberbach ldquoUber einige extremal problem in gebiete derkonformen abbildungrdquo Mathematische Annalen vol 77 no 2pp 153ndash172 1916

[22] T Hayami S Owa and H M Srivastava ldquoCoefficient inequal-ities for certain classes of analytic and univalent functionsrdquoJournal of Inequalities in Pure and Applied Mathematics vol 8no 4 pp 1ndash10 2007

[23] M S Liu and Y C Zhu ldquoGeneralized Roper-Suffridge oper-ators on bounded and complete Reinhardt domainsrdquo Sciencein China vol 37 no 10 pp 1193ndash1206 2007

[24] Y Y Cui and C J Wang ldquoThe generalized Roper-Suffridgeextension operators in complex Banach spacerdquo Journal ofMathematics vol 34 no 3 pp 515ndash520 2014

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Stochastic AnalysisInternational Journal of

4 Discrete Dynamics in Nature and Society

where119875 = (2119898119909 + 2 |1 minus 120582|) (|1 minus 120582|2 minus |1 + 120582|2 1199092) + (1198981199092+ 2 |1 minus 120582| 119909 + |1 minus 120582|2) 2 |1 + 120582|2 119909 = 2 |1 minus 120582| |1+ 120582|2 1199092 + 2 |1 minus 120582|2 (119898 + |1 + 120582|2) 119909 + 2 |1 minus 120582|3= 2 |1 minus 120582| [|1 + 120582|2 1199092 + |1 minus 120582| |1 + 120582|2sdot 2 (1 +R120582)1 + |120582|2 + 2R120582119909 + |1 minus 120582|2] = 2 |1 minus 120582| |1 + 120582|2sdot [[(119909 + |1 minus 120582| (1 +R120582)1 + |120582|2 + 2R120582 )2 + |1 minus 120582|2|1 + 120582|2minus |1 minus 120582|2 (1 +R120582)2(1 + |120582|2 + 2R120582)2]] = 2 |1 minus 120582| |1 + 120582|2(119909+ |1 minus 120582| (1 +R120582)1 + |120582|2 + 2R120582 )2+ |1 minus 120582|2|1 + 120582|2 (1 + |120582|2 + 2R120582)2 [(1 + |120582|2 + 2R120582)2minus |1 + 120582|2 (1 +R120582)2] = 2 |1 minus 120582| |1 + 120582|2 [[(119909+ |1 minus 120582| (1 +R120582)1 + |120582|2 + 2R120582 )2+ |1 minus 120582|2 (I120582)2|1 + 120582|2 (1 + |120582|2 + 2R120582)2]] ge 0

(24)

which follows 1198911015840(119909) ge 0 Therefore 119891(119909) is monotoneincreasing So

R119886 + 120588 = 119898 |119911|2 + 2 |1 minus 120582| |119911| + |1 minus 120582|2|1 minus 120582|2 minus |1 + 120582|2 |119911|2le 1198981199032 + 2 |1 minus 120582| 119903 + |1 minus 120582|2|1 minus 120582|2 minus |1 + 120582|2 1199032 (25)

where 119911 = 119903119890119894120579 120579 isin [0 2120587) By (19) we haveR(1199111198911015840(119911)119891(119911)) =119903(120597120597119903) ln |119891(119911)| le R119886 + 120588 Therefore120597120597119903 ln 1003816100381610038161003816119891 (119911)1003816100381610038161003816 le 1119903 1198981199032 + 2 |1 minus 120582| 119903 + |1 minus 120582|2|1 minus 120582|2 minus |1 + 120582|2 1199032= 1119903 (|1 minus 120582|2 minus |1 + 120582|2 1199032) [ |120582|2 minus 11 + |120582|2 + 2R120582 (|1minus 120582|2 minus |1 + 120582|2 1199032) + 1 minus |120582|21 + |120582|2 + 2R120582 |1 minus 120582|2 + |1

minus 120582|2 + 2 |1 minus 120582| 119903] = 1119903 [ |120582|2 minus 11 + |120582|2 + 2R120582 + 2 |1minus 120582| 119903 + (1 +R120582) |1 minus 120582| (1 + |120582|2 + 2R120582)|1 minus 120582|2 minus |1 + 120582|2 1199032 ]

(26)

Integrating both ends of (26) on [120576 119903] we obtainint119903120576

120597120597119905 ln 10038161003816100381610038161003816119891 (119905119890119894120579)10038161003816100381610038161003816 119889119905 le |120582|2 minus 11 + |120582|2 + 2R120582 int119903120576

119889119905119905 + 2 |1minus 120582| [int119903

120576

119889119905|1 minus 120582|2 minus |1 + 120582|2 1199052+ (1 +R120582) |1 minus 120582|1 + |120582|2 + 2R120582 int119903

120576

1119905 119889119905|1 minus 120582|2 minus |1 + 120582|2 1199052]= |120582|2 minus 11 + |120582|2 + 2R120582 ln 119905|119903120576 + 2 |1 minus 120582| 12 10038161003816100381610038161 minus 12058221003816100381610038161003816sdot ln 10038161003816100381610038161003816100381610038161003816 119905 + |(1 minus 120582) (1 + 120582)|119905 minus |(1 minus 120582) (1 + 120582)| 1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119903120576 + 2 (1 +R120582) |1 minus 120582|21 + |120582|2 + 2R120582sdot 12 |1 minus 120582|2 ln 119905210038161003816100381610038161003816|1 minus 120582|2 minus |1 + 120582|2 119905210038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816100381610038161003816119903

120576

(27)

Therefore

ln 1003816100381610038161003816119891 (119911)1003816100381610038161003816 minus ln 10038161003816100381610038161003816119891 (120576119890119894120579)10038161003816100381610038161003816 le |120582|2 minus 11 + |120582|2 + 2R120582 (ln 119903minus ln 120576) + 1|1 + 120582| (ln 10038161003816100381610038161003816100381610038161003816 119903 + |(1 minus 120582) (1 + 120582)|119903 minus |(1 minus 120582) (1 + 120582)| 10038161003816100381610038161003816100381610038161003816minus ln

10038161003816100381610038161003816100381610038161003816 120576 + |(1 minus 120582) (1 + 120582)|120576 minus |(1 minus 120582) (1 + 120582)| 10038161003816100381610038161003816100381610038161003816)+ (1 +R120582)1 + |120582|2 + 2R120582 (ln 119903210038161003816100381610038161003816|1 minus 120582|2 minus |1 + 120582|2 119903210038161003816100381610038161003816minus ln 120576210038161003816100381610038161003816|1 minus 120582|2 minus |1 + 120582|2 120576210038161003816100381610038161003816)

(28)

Since

ln 10038161003816100381610038161003816119891 (120576119890119894120579)10038161003816100381610038161003816 minus |120582|2 minus 11 + |120582|2 + 2R120582 ln 120576minus 1 +R1205821 + |120582|2 + 2R120582 ln 1205762 = ln

10038161003816100381610038161003816119891 (120576119890119894120579)10038161003816100381610038161003816120576 997888rarr 0(120576 997888rarr 0)

(29)

Discrete Dynamics in Nature and Society 5

setting 120576 rarr 0 in (28) we have

ln 1003816100381610038161003816119891 (119911)1003816100381610038161003816 le |120582|2 minus 11 + |120582|2 + 2R120582 ln 119903 + 1|1 + 120582|sdot ln 10038161003816100381610038161003816100381610038161003816 119903 + |(1 minus 120582) (1 + 120582)|119903 minus |(1 minus 120582) (1 + 120582)| 10038161003816100381610038161003816100381610038161003816+ 1 +R1205821 + |120582|2 + 2R120582 (ln 119903210038161003816100381610038161003816|1 minus 120582|2 minus |1 + 120582|2 119903210038161003816100381610038161003816+ ln |1 minus 120582|2)

(30)

Therefore1003816100381610038161003816119891 (119911)1003816100381610038161003816 le 119903(|120582|2minus1)(1+|120582|2+2R120582)sdot 10038161003816100381610038161003816100381610038161003816 119903 + |(1 minus 120582) (1 + 120582)|119903 minus |(1 minus 120582) (1 + 120582)| 100381610038161003816100381610038161003816100381610038161|1+120582|sdot ( |1 minus 120582|2 119903210038161003816100381610038161003816|1 minus 120582|2 minus |1 + 120582|2 119903210038161003816100381610038161003816)

(1+R120582)(1+|120582|2+2R120582)

= 119903 10038161003816100381610038161003816100381610038161003816 119903 + |(1 minus 120582) (1 + 120582)|119903 minus |(1 minus 120582) (1 + 120582)| 100381610038161003816100381610038161003816100381610038161|1+120582|sdot ( 1100381610038161003816100381610038161 minus |(1 + 120582) (1 minus 120582)|2 119903210038161003816100381610038161003816)

(1+R120582)(1+|120582|2+2R120582)

= 119903 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)(1+|120582|2+2R120582)sdot 10038161003816100381610038161003816100381610038161003816119903 minus 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 1205821003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816minus1|1+120582|minus(1+R120582)(1+|120582|2+2R120582)sdot 10038161003816100381610038161003816100381610038161003816119903 + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161|1+120582|minus(1+R120582)(1+|120582|2+2R120582)= 119903 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2sdot 10038161003816100381610038161003816100381610038161003816119903 minus 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 1205821003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(minus|1+120582|minus1minusR120582)|1+120582|2sdot 10038161003816100381610038161003816100381610038161003816119903 + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 1205821003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(|1+120582|minus1minusR120582)|1+120582|2 = |119911|sdot 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2sdot 10038161003816100381610038161003816100381610038161003816|119911| minus 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 1205821003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(minus|1+120582|minus1minusR120582)|1+120582|2sdot (|119911| + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816)(|1+120582|minus1minusR120582)|1+120582|2

(31)

Obviously |(1 minus 120582)(1 + 120582)| ge 1 forR120582 le 0 Thus |119911| le |(1 minus120582)(1 + 120582)| So we obtain1003816100381610038161003816119891 (119911)1003816100381610038161003816 le |119911| 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2

sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 minus |119911|)(minus|1+120582|minus1minusR120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 + |119911|)(|1+120582|minus1minusR120582)|1+120582|2

(32)

Similarly by (19) we can get

1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge |119911| 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 minus |119911|)(|1+120582|minus1minusR120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 + |119911|)(minus|1+120582|minus1minusR120582)|1+120582|2

(33)

Theorem8 Let119891(119911) be an almost starlike function of complexorder 120582 on119863 with 120582 = minus1 Then

1 minus |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le 1 + |119911| |119911| 119890minus|119911| le 1003816100381610038161003816119891 (119911)1003816100381610038161003816 le |119911| 119890|119911|(1 minus |119911|) 119890minus|119911| le 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 le (1 + |119911|) 119890|119911|

(34)

Proof By (14) we have |2119892(119911) minus 1| le |119911| which follows|1199111198911015840(119911)119891(119911) minus 1| le |119911| Therefore

1 minus |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le 1 + |119911| 1 minus |119911| le R

1199111198911015840 (119911)119891 (119911) le 1 + |119911| (35)

For |119911| = 119903 we have1 minus 119903 le 119903 120597120597119903 ln 1003816100381610038161003816119891 (119911)1003816100381610038161003816 = R

1199111198911015840 (119911)119891 (119911) le 1 + 119903 (36)

Similar to Theorem 7 we get

int119903120576

119889119905119905 minus int119903120576119889119905 le int119903

120576

120597120597119905 ln 10038161003816100381610038161003816119891 (119905119890119894120579)10038161003816100381610038161003816 119889119905le int119903120576

119889119905119905 + int119903120576119889119905 (37)

6 Discrete Dynamics in Nature and Society

Let 120576 rarr 0 we obtain |119911|119890minus|119911| le |119891(119911)| le |119911|119890|119911| Hence (1 minus|119911|)119890minus|119911| le |1198911015840(119911)| le (1 + |119911|)119890|119911|Setting 120582 = 120572(120572 minus 1) in Theorems 7 and 8 we get the

following corollary

Corollary 9 Let 119891(119911) be an almost starlike function of order120572 Then |119911|(1 + (1 minus 2120572) |119911|)2(1minus120572)(1minus2120572) le 1003816100381610038161003816119891 (119911)1003816100381610038161003816le |119911|(1 minus (1 minus 2120572) |119911|)2(1minus120572)(1minus2120572) 120572 isin [0 1) 12

|119911| 119890minus|119911| le 1003816100381610038161003816119891 (119911)1003816100381610038161003816 le |119911| 119890|119911| 120572 = 12 1 minus |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le 1 + |119911| (1 minus |119911|) 119890minus|119911| le 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 le (1 + |119911|) 119890|119911|

120572 = 12

(38)

Theorem 10 Let 119891(119911) be an almost starlike function ofcomplex order 120582 on119863 with 120582 isin C minus1 Re 120582 le 0 Then1 minus |119911|1 + |(1 + 120582) (1 minus 120582)| |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816le 1 + |119911|1 minus |(1 + 120582) (1 minus 120582)| |119911|

(1 minus |119911|) 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2+1sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 minus |119911|)(|1+120582|minus1minusR120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 + |119911|)(minus|1+120582|minus1minusR120582)|1+120582|2minus1 le 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816le (1 + |119911|) 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2+1sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 + |119911|)(|1+120582|minus1minusR120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 minus |119911|)(minus|1+120582|minus1minusR120582)|1+120582|2minus1

(39)

Proof By (17) we have100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) minus 1100381610038161003816100381610038161003816100381610038161003816 le 10038161003816100381610038161003816100381610038161003816 1 + 1205821 minus 12058210038161003816100381610038161003816100381610038161003816 |119911| 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) + 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038161003816 (40)

Since 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 minus 1 le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) minus 1100381610038161003816100381610038161003816100381610038161003816 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) + 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038161003816 le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 (41)

then we have100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 minus 1 le (100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816) 10038161003816100381610038161003816100381610038161003816 1 + 1205821 minus 12058210038161003816100381610038161003816100381610038161003816 |119911| (42)

which follows100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le 1 + |119911|1 minus |(1 + 120582) (1 minus 120582)| |119911| (43)

On the other hand we also have

1 minus 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le (100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816) 10038161003816100381610038161003816100381610038161003816 1 + 1205821 minus 12058210038161003816100381610038161003816100381610038161003816 |119911| (44)

which leads to100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 ge 1 minus |119911|1 + |(1 + 120582) (1 minus 120582)| |119911| (45)

Hence 1 minus |119911|1 + |(1 + 120582) (1 minus 120582)| |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816le 1 + |119911|1 minus |(1 + 120582) (1 minus 120582)| |119911| (46)

Then by Theorem 7 we obtain the desired conclusion (39)

Setting 120582 = 120572(120572 minus 1) inTheorem 10 we get the followingresult

Corollary 11 Let 119891(119911) be an almost starlike function of order120572 with 120572 isin [0 1) 12 Then1 minus |119911|1 + |1 minus 2120572| |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le 1 + |119911|1 minus |1 minus 2120572| |119911| 1 minus |119911|1 + |1 minus 2120572| |119911| |119911|(1 + (1 minus 2120572) |119911|)2(1minus120572)(1minus2120572)le 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816le 1 + |119911|1 minus |1 minus 2120572| |119911| |119911|(1 minus (1 minus 2120572) |119911|)2(1minus120572)(1minus2120572) (47)

3 Coefficient Estimates of Almost StarlikeFunctions of Complex Order 120582

Lemma 12 (see [21]) Let 119901(119911) = 1 + suminfin119899=1 119888119899119911119899 be aholomorphic function on 119863 with R119901(119911) gt 0 Then |119888119899| le2 (119899 = 1 2 )

Discrete Dynamics in Nature and Society 7

Lemma 13 (see [22]) Let119901(119911) = 1+suminfin119899=1 119888119899119911119899 be holomorphicon119863 ThenR119901(119911) gt 0 if and only if

119901 (119911) = 120585 minus 1120585 + 1 120585 isin 120597119863 (48)

Theorem 14 Let 119891(119911) = 119911 + suminfin119899=2 119886119899119911119899 be an almost starlikefunction of complex order 120582 with 120582 isin C Re 120582 le 0 on119863 Then10038161003816100381610038161198861198991003816100381610038161003816 le 119899minus1prod

119895=2

( 2|1 minus 120582| + 119895 minus 2119895 minus 1) 119899 = 2 3 (49)

Proof Since 119891(119911) is an almost starlike function of complexorder 120582 then

R [(1 minus 120582) 119891 (119911)1199111198911015840 (119911) + 120582] gt 0 119911 isin 119863 (50)

Let 119901(119911) = (1 minus 120582)(119891(119911)1199111198911015840(119911)) + 120582 Then we have1199111198911015840 (119911) (119901 (119911) minus 120582) = (1 minus 120582) 119891 (119911) (51)

and 119901(0) = 1 andR119901(119911) gt 0 Let 119901(119911) = 1 + suminfin119899=1 119888119899119911119899 Then10038161003816100381610038161198881198991003816100381610038161003816 le 2 (119899 = 1 2 ) (52)

by Lemma 12 For 119891(119911) = 119911 + suminfin119899=2 119886119899119911119899 (51) follows119911(1 + infinsum

119899=2

119899119886119899119911119899minus1)(1 minus 120582 + infinsum119899=1

119888119899119911119899)= (1 minus 120582)(119911 + infinsum

119899=2

119886119899119911119899) (53)

Therefore(1 minus 120582) 119911 + [1198881 + 21198862 (1 minus 120582)] 1199112 + sdot sdot sdot + [119888119899minus1+ 21198862119888119899minus2 + sdot sdot sdot + (119899 minus 1) 119886119899minus11198881 + 119899119886119899 (1 minus 120582)] 119911119899+ sdot sdot sdot = (1 minus 120582) 119911 + 1198862 (1 minus 120582) 1199112 + sdot sdot sdot + 119886119899 (1 minus 120582)sdot 119911119899 + sdot sdot sdot (54)

So 119886119899 = 119888119899minus1 + 21198862119888119899minus2 + sdot sdot sdot + (119899 minus 1) 119886119899minus11198881(119899 minus 1) (120582 minus 1) (55)

By (52) we get10038161003816100381610038161198861198991003816100381610038161003816 le 2 [1 + 2 100381610038161003816100381611988621003816100381610038161003816 + sdot sdot sdot + (119899 minus 1) 1003816100381610038161003816119886119899minus11003816100381610038161003816](119899 minus 1) |1 minus 120582| (56)

Bymathematical induction we obtain the desired conclusion

Theorem 15 Let 119891(119911) be an almost starlike function ofcomplex order 120582 with 120582 isin C Re 120582 le 0 on 119863 and let 119911 = 0be the zero of order 119896 + 1 (119896 isin N) of 119891(119911) minus 119911 Let 119891(119911) =119911 + suminfin119899=119896+1 119886119899119911119899 Then1003816100381610038161003816119886119899+11003816100381610038161003816 le 2|120582 + 119899| (119899 = 119896 119896 + 1 ) (57)

Proof Since 119891(119911) = 119911 + suminfin119899=119896+1 119886119899119911119899 similar to Theorem 14by (51) we obtain

119911(1 + infinsum119899=119896+1

119899119886119899119911119899minus1)(1 minus 120582 + infinsum119899=1

119888119899119911119899)= (1 minus 120582)(119911 + infinsum

119899=119896+1

119886119899119911119899) (58)

which follows(1 minus 120582) 119911 + 11988811199112 + sdot sdot sdot + 119888119896minus1119911119896+ [119888119896 + (119896 + 1) 119886119896+1] 119911119896+1 + sdot sdot sdot+ [119888119899 + (119899 + 1) 119886119899+1] 119911119899+1 + sdot sdot sdot= (1 minus 120582) 119911 + (1 minus 120582) 119886119896+1119911119896+1 + sdot sdot sdot+ (1 minus 120582) 119886119899+1119911119899+1 + sdot sdot sdot (59)

Therefore 1198881 = sdot sdot sdot = 119888119896minus1 = 0119888119899 = (minus120582 minus 119899) 119886119899+1 (119899 = 119896 119896 + 1 ) (60)

By (52) we get the desired conclusion

Theorem 16 119891(119911) = 119911 + suminfin119899=2 119886119899119911119899 is an almost starlikefunction of complex order 120582 with 120582 isin C and Re 120582 le 0 on 119863if and only ifinfinsum119899=2

[(1 + (119899 minus 1) 120582) (120585 + 1) + 119899 (1 minus 120585)] 119886119899119911119899 = minus2(120585 isin 120597119863 119899 = 2 3 ) (61)

Proof Let 119901(119911) = (1 minus 120582)(119891(119911)1199111198911015840(119911)) + 120582 Since 119891(119911) isan almost starlike function of complex order 120582 then 119901(119911)is holomorphic on 119863 and 119901(0) = 0 and R119901(119911) gt 0 ByLemma 13 we get(1 minus 120582) 119891 (119911)1199111198911015840 (119911) + 120582 = 120585 minus 1120585 + 1 120585 isin 120597119863 (62)

which follows(1 minus 120582) (120585 + 1) 119891 (119911) + 120582 (120585 + 1) 1199111198911015840 (119911)= (120585 minus 1) 1199111198911015840 (119911) (63)

For 119891(119911) = 119911 + suminfin119899=2 119886119899119911119899 we have(1 minus 120582) (120585 + 1) (119911 + infinsum119899=2

119886119899119911119899)+ 120582 (120585 + 1) (119911 + infinsum

119899=2

119899119886119899119911119899)= (120585 minus 1) (119911 + infinsum

119899=2

119899119886119899119911119899) (64)

which leads to the desired conclusion

8 Discrete Dynamics in Nature and Society

Remark 17 Setting 120582 = 120572(120572 minus 1) in Theorems 14ndash16 weget the corresponding results for almost starlike functions oforder 1205724 The Invariance of Almost Starlike Mappings

of Complex Order 120582In this section we mainly discuss the invariance of almoststarlike mappings of complex order 120582 on Reinhardt domainsand on the unit ball in complex Banach spaces under somegeneralized Roper-Suffridge operators

Lemma 18 (see [23]) Let Ω sub C119899 be a bounded andcompletely Reinhardt domain whose Minkowski functional120588(119911) is 1198621 except for a lower-dimensional manifold Ω0 Thenfor forall119911 = (1199111 119911119899) isin Ω Ω0 we have120597120588 (119911)120597119911 = (120597120588 (119911)1205971199111 120597120588 (119911)1205971199112 120597120588 (119911)120597119911119899 )

120597120588 (119911)120597119911119895 119911119895 ge 0120588 (119911) = 2 119899sum

119895=1

120597120588 (119911)120597119911119895 119911119895(65)

Theorem 19 Let 119891(119911) be an almost starlike function of com-plex order 120582 on 119863 with 120582 isin C and Re 120582 le 0 Let Ω sub C119899 be abounded and completely Reinhardt domain whose Minkowskifunctional 120588(119911) is 1198621 on Ω 0 Let

119865 (119911) = (119903119891(1199111119903 ) (119903119891 (1199111119903)1199111 )1205732sdot 1199112 (119903119891 (1199111119903)1199111 )120573119899 119911119899)1015840

(66)

where 120573119895 isin [0 1] 119903 = sup|1199111| 119911 = (1199111 119911119899)1015840 isin Ω and(119903119891(1199111119903)1199111)120573119895 |1199111=0 = 1 (119895 = 2 119899) Then 119865(119911) is an almoststarlike mapping of complex order 120582 on ΩProof For

119865 (119911) = (119903119891(1199111119903 ) (119903119891 (1199111119903)1199111 )1205732sdot 1199112 (119903119891 (1199111119903)1199111 )120573119899 119911119899)1015840

(67)

immediately we have

119869119865 (119911) =(((((((((((

1198911015840 (1199111119903 ) 0 sdot sdot sdot 01205732119903 [1198911015840 (1199111119903)119891 (1199111119903) minus 1199031199111 ](119903119891 (1199111119903)1199111 )1205732 1199112 (119903119891 (1199111119903)1199111 )1205732 sdot sdot sdot 0

sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot120573119899119903 [1198911015840 (1199111119903)119891 (1199111119903) minus 1199031199111](119903119891 (1199111119903)1199111 )120573119899 119911119899 0 sdot sdot sdot (119903119891 (1199111119903)1199111 )120573119899

)))))))))))

(68)

Therefore

119869minus1119865 (119911) =(((((((((((

11198911015840 (1199111119903) 0 sdot sdot sdot 01205732119903 [ 11990311991111198911015840 (1199111119903) minus 1119891 (1199111119903)] 1199112 (119903119891 (1199111119903)1199111 )minus1205732 sdot sdot sdot 0

sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot120573119899119903 [ 11990311991111198911015840 (1199111119903) minus 1119891 (1199111119903)] 119911119899 0 sdot sdot sdot (119903119891 (1199111119903)1199111 )minus120573119899

)))))))))))

(69)

Discrete Dynamics in Nature and Society 9

Then 119869minus1119865 (119911) 119865 (119911)= (119903119891 (1199111119903)1198911015840 (1199111119903) (1 minus 1205732 + 1205732 119903119891 (1199111119903)11991111198911015840 (1199111119903))sdot 1199112 (1 minus 120573119899 + 120573119899 119903119891 (1199111119903)11991111198911015840 (1199111119903)) 119911119899)1015840

(70)

By Lemma 18 we get(1 minus 120582) 2120597120588120597119911 (119911) 119869minus1119865 (119911) 119865 (119911) = 2 (1 minus 120582)sdot [[ 119903119891 (1199111119903)11991111198911015840 (1199111119903) 120597120588 (119911)1205971199111 1199111+ 119899sum119895=2

(1 minus 120573119895 + 120573119895 119903119891 (1199111119903)11991111198911015840 (1199111119903)) 120597120588 (119911)120597119911119895 119911119895]] (71)

Since119891(119911) is an almost starlike function of complex order120582 thenR[(1 minus 120582) 119903119891 (1199111119903)11991111198911015840 (1199111119903)] ge minusR120582 (72)

Therefore by Lemma 18 we obtain

R [(1 minus 120582) 2120597120588120597119911 (119911) 119869minus1119865 (119911) 119865 (119911)] = R[(1 minus 120582)sdot 119903119891 (1199111119903)11991111198911015840 (1199111119903)] 2120597120588 (119911)1205971199111 1199111 + 119899sum

119895=2

(1 minus 120573119895)R (1 minus 120582)+ 120573119895R[(1 minus 120582) 119903119891 (1199111119903)11991111198911015840 (1199111119903)] 2120597120588 (119911)120597119911119895 119911119895 ge minusR120582sdot 2120597120588 (119911)1205971199111 1199111 + 119899sum

119895=2

(1 minus 120573119895) (1 minusR120582) + 120573119895 (minusR120582)sdot 2120597120588 (119911)120597119911119895 119911119895 = (minusR120582) [2120597120588 (119911)1205971199111 1199111 + 2120597120588 (119911)120597119911119895 119911119895]+ 119899sum119895=2

(1 minus 120573119895) 2120597120588 (119911)120597119911119895 119911119895 ge (minusR120582) 120588 (119911)

(73)

Hence 119865(119911) is an almost starlike mapping of complex order 120582onΩ by Definition 4

Theorem 20 Let 119891119895 (119895 = 1 2 119899) be almost starlikefunctions of complex order 120582 on 119863 with 120582 isin C and Re 120582 lt 0Let Ω sub C119899 be a bounded starlike and circular domain whoseMinkowski functional 120588(119911) is1198621 onΩ0 Let 119903119895 be the radiusof the disk 119880119895 = 119911119895 isin C 119911 = (1199111 119911119895 119911119899)1015840 isin Ω whosecenter is zero Let

119865 (119911) = 119911 119899prod119895=1

(119903119895119891119895 (119911119895119903119895)119911119895 )120582119894 (74)

where 120582119894119895 ge 0 sum119899119895=1 120582119894119895 = 1 and (119891119895(119911119895)119911119895)120582119894119895 |119911119895=0 = 1 (119894 119895 =1 2 119899)Then119865(119911) is an almost starlikemapping of complexorder 120582 on ΩProof Let

120577119895 = 119911119895119903119895 119892 (119911) = 119899prod

119895=1

(119891119895 (120577119895)120577119895 )120582119894 (75)

which follows 119865(119911) = 119911119892(119911) Then 119869119865(119911) = 119869119892(119911)119911 + 119892(119911) Bydirect computation we get

119869119892 (119911) 119911 = 119892 (119911) [[119899sum119895=1

1205821198951198911015840119895 (119911119895119903119895) 119911119895119903119895119891119895 (119911119895119903119895) minus 1]]= 119892 (119911) [[

119899sum119895=1

1205821198951198911015840119895 (120577119895) 120577119895119891119895 (120577119895) minus 1]] (76)

Since 119891119895 are almost starlike functions of complex order 120582 on119863 then

R[[(1 minus 120582) 119891119895 (120577119895)1205771198951198911015840119895 (120577119895)]] ge minusR120582 gt 0 (77)

Therefore

R[ 11 minus 120582 (1 + 119869119892 (119911) 119911119892 (119911) )]= R[[ 11 minus 120582 119899sum

119895=1

1205821198951198911015840119895 (120577119895) 120577119895119891119895 (120577119895) ]]= 119899sum119895=1

120582119895R[[ 11 minus 120582 1198911015840119895 (120577119895) 120577119895119891119895 (120577119895) ]] gt 0(78)

Then 119869119892(119911)119911+119892(119911) = 0 It is obvious that119865(0) = 0 and 119869119865(0) =119868 so 119865(119911) is a normalized locally biholomorphic mapping onΩ and

119869minus1119865 (119911) 119865 (119911) = 119892 (119911) 119911119869119892 (119911) 119911 + 119892 (119911) (79)

Let

119902119895 (120577119895) = 2Re (minus120582) 1198911015840119895 (120577119895) 120577119895(1 minus 120582) 119891119895 (120577119895) minus 1 (80)

(77) follows 100381610038161003816100381610038161003816100381610038161003816100381610038162Re (minus120582) 1198911015840119895 (120577119895) 120577119895(1 minus 120582) 119891119895 (120577119895) minus 110038161003816100381610038161003816100381610038161003816100381610038161003816 le 1 (81)

10 Discrete Dynamics in Nature and Society

that is |119902119895(120577119895)| le 1 For Re 120582 lt 0 by (76) (79) and Lemma 18we obtain1003816100381610038161003816100381610038161003816100381610038162R (minus120582) 1(1 minus 120582) (2120588 (119911)) (120597120588120597119911) (119911) 119869minus1119865 (119911) 119865 (119911)

minus 1100381610038161003816100381610038161003816100381610038161003816 =100381610038161003816100381610038161003816100381610038161003816100381610038162R (minus120582) 11 minus 120582 119899sum

119895=1

1205821198951198911015840119895 (120577119895) 120577119895119891119895 (120577119895) minus 110038161003816100381610038161003816100381610038161003816100381610038161003816= 10038161003816100381610038161003816100381610038161003816100381610038161003816 119899sum119895=1120582119895119902119895 (120577119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816 le 119899sum119895=1120582119895 = 1(82)

which leads to

R [(1 minus 120582) 2120597120588120597119911 (119911) 119869minus1119865 (119911) 119865 (119911)] ge minusR120582120588 (119911) (83)

Hence 119865(119911) is an almost starlike mapping of complex order 120582onΩ by Definition 4

Similar to Theorem 20 we can get the following conclu-sion

Theorem 21 Let 119891119895 (119895 = 1 2 119899) be almost starlikefunctions of complex order 120582 on 119863 with 120582 isin C and Re 120582 lt 0Let

119865 (119909) = 119909 119899prod119895=1

(119891119895 (119879119906119895 (119909))119879119906119895 (119909) )120582119895 119909 isin 119861 (84)

where 120582119894119895 ge 0 sum119899119895=1 120582119894119895 = 1 (119891119895(119909119895)119909119895)120582119894119895 |119909119895=0 = 1 (119909119895 =119879119906119895(119909) 119894 119895 = 1 2 119899) and 119906119895 is the unit vector of thecomplex Banach space 119883 Then 119865(119909) is an almost starlikemapping of complex order 120582 on the unit ball 119861 in complexBanach space119883

Theorem 22 Let 119891 be an almost starlike function of complexorder 120582 on119863 with 120582 isin C and Re 120582 le 0 Let

119865 (119909) = 119899minus1sum119895=1

(119891(1198791199091 (119909))1198791199091 (119909) )120573119895 119879119909119895 (119909) 119909119895 + 119909minus 119899minus1sum119895=1

119879119909119895 (119909) 119909119895 119909 isin 119861 (85)

where 1199091 isin 119861 1199091 = 1 120573119895 isin [0 1] (119891(119911)119911)120573119895 |119911=0 = 1 (119895 =1 2 119899 minus 1) and 1199091 119909119899 are linearly independent and119879119909119894 isin 119883lowast such that 119879119909119894 = 1 119879119909119894(119909119894) = 1 and 119879119909119894(119909119895) =0 (119894 = 119895) for forall119909119894 Then 119865(119909) is an almost starlike mapping ofcomplex order 120582 on the unit ball 119861 in complex Banach space119883

Proof From [24] we get 119865(119909) is a normalized biholomorphicmapping on 119861 and119909 119879119909 [119863119865 (119909)minus1 119865 (119909)]= 1199092

+ 119899minus1sum119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162 [ 119891 (1198791199091 (119909))1198791199091 (119909) 1198911015840 (1198791199091 (119909)) minus 1] (86)

Since119891 is an almost starlike function of complex order120582 thenR[[(1 minus 120582) 119891 (119879119909119895 (119909))119879119909119895 (119909) 1198911015840 (119879119909119895 (119909))]] ge minusR120582 (87)

Therefore

R (1 minus 120582) 119879119909 [119863119865 (119909)minus1 119865 (119909)] = (1 minusR120582) 119909+ 119899minus1sum119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162119909sdotR(1 minus 120582) [ 119891 (1198791199091 (119909))1198791199091 (119909) 1198911015840 (1198791199091 (119909)) minus 1]ge (1 minusR120582) 119909 + 119899minus1sum

119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162119909sdot [R (minus120582) minus (1 minusR120582)] = minusR120582 119909+ 1119909 [[1199092 minus

119899minus1sum119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162]] ge minusR120582 119909

(88)

which lead to the desired conclusion by Definition 5

Similar to Theorem 22 we can get the following conclu-sion

Theorem 23 Let 119891 be an almost starlike function of complexorder 120582 on119863 with 120582 isin C Re 120582 le 0 Let119865 (119909) = Φ120573 (119891) (119909)= 119891 (1198791199091 (119909)) 1199091

+ (119891 (1198791199091 (119909))1198791199091 (119909) )120573 (119909 minus 1198791199091 (119909) 1199091) (89)

where 0 le 120573 le 1 1199091 isin 119861 1199091 = 1 (119891(1199111)1199111)120573|1199111=0 = 1 and119891(1199111) = 0 for 1199111 = 0 Then 119865(119909) is an almost starlike mappingof complex order 120582 on 119861Remark 24 Setting 120582 = 120572(120572 minus 1) in Theorems 19ndash23 weget the corresponding results with respect to almost starlikemappings of order 120572

Discrete Dynamics in Nature and Society 11

Competing Interests

The authors declare that they have no conflict of interests

Acknowledgments

This work is supported by NSF of China (no U1204618) andScience and Technology Research Projects of Henan Provin-cial Education Department (nos 14B110015 and 14B110016)

References

[1] C Pommerenke Univalent Functions Vandenhoeck amp Ru-precht Gottingen Germany 1975

[2] H Cartan ldquoSur la possibilite drsquoentendre aux functions de plus-ieurs variables complexes la theorie des functions univalentsrdquoin Lecons sur les Functions Univalents onMutivalents P MontelEd Gauthier-Villar Paris France 1933

[3] R W Barnard C H FitzGerald and S Gong ldquoA distortionthoerem for biholomorphicmappings inC2rdquoTransactions of theAmerican Mathematical Society vol 344 pp 902ndash924 1994

[4] T S Liu andW J Zhang ldquoA distortion thoerem for biholomor-phic convex mappings in C119899rdquo Chinese Journal of ContemporaryMathematics vol 20 no 3 pp 421ndash430 1999

[5] S Gong S K Wang and Q H Yu ldquoBiholomorphic convexmappings of ball in C119899rdquo Pacific Journal of Mathematics vol 161no 2 pp 287ndash306 1993

[6] S Gong and T S Liu ldquoDistortion theorems for biholomorphicconvex mappings on bounded convex circular domainsrdquo Chi-nese Annals of Mathematics Series B vol 20 no 3 pp 297ndash3041999

[7] I Graham and D Varolin ldquoBloch constants in one and severalvariablesrdquoPacific Journal ofMathematics vol 174 no 2 pp 347ndash357 1996

[8] P L DurenUnivalent Functions Springer New York NY USA1983

[9] T H Gronwall ldquoSome remarks on conformal representationrdquoAnnals of Mathematics vol 16 no 1ndash4 pp 72ndash76 191415

[10] M I Robertson ldquoOn the theory of univalent functionsrdquo Annalsof Mathematics vol 37 no 2 pp 374ndash408 1936

[11] A V Boyd ldquoCoefficient estimates for starlike functions of order120572rdquo Proceedings of the AmericanMathematical Society vol 17 pp1016ndash1018 1966

[12] H Silverman ldquoUnivalent functions with negative coefficientsrdquoProceedings of the American Mathematical Society vol 51 pp109ndash116 1975

[13] H M Srivastava and S Owa ldquoCertain subclasses of starlikefunctions Irdquo Journal ofMathematical Analysis and Applicationsvol 161 no 2 pp 405ndash415 1991

[14] K A Roper and T J Suffridge ldquoConvex mappings on the unitball ofC119899rdquo Journal drsquoAnalyseMathematique vol 65 pp 333ndash3471995

[15] I Graham and G Kohr ldquoUnivalent mappings associated withthe Roper-Suffridge extension operatorrdquo Journal drsquoAnalyseMathematique vol 81 pp 331ndash342 2000

[16] S Gong and T S Liu ldquoThe generalized Roper-Suffridge exten-sion operatorrdquo Journal of Mathematical Analysis and Applica-tions vol 284 no 2 pp 425ndash434 2003

[17] G Kohr ldquoLoewner chains and a modification of the Roper-Suffridge extension operatorrdquo Mathematica vol 71 no 1 pp41ndash48 2006

[18] J R Muir ldquoA class of Loewner chain preserving extensionoperatorsrdquo Journal of Mathematical Analysis and Applicationsvol 337 no 2 pp 862ndash879 2008

[19] Y H Zhao Almost Starlike Mappings of Complex Order 120582 onthe Unit Ball of a Complex Banach Space Zhejiang NormalUniversity Jinhua China 2013

[20] L V Ahlfors Complex Analysis McGraw-Hill New York NYUSA 3rd edition 1978

[21] L Bieberbach ldquoUber einige extremal problem in gebiete derkonformen abbildungrdquo Mathematische Annalen vol 77 no 2pp 153ndash172 1916

[22] T Hayami S Owa and H M Srivastava ldquoCoefficient inequal-ities for certain classes of analytic and univalent functionsrdquoJournal of Inequalities in Pure and Applied Mathematics vol 8no 4 pp 1ndash10 2007

[23] M S Liu and Y C Zhu ldquoGeneralized Roper-Suffridge oper-ators on bounded and complete Reinhardt domainsrdquo Sciencein China vol 37 no 10 pp 1193ndash1206 2007

[24] Y Y Cui and C J Wang ldquoThe generalized Roper-Suffridgeextension operators in complex Banach spacerdquo Journal ofMathematics vol 34 no 3 pp 515ndash520 2014

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Discrete Dynamics in Nature and Society 5

setting 120576 rarr 0 in (28) we have

ln 1003816100381610038161003816119891 (119911)1003816100381610038161003816 le |120582|2 minus 11 + |120582|2 + 2R120582 ln 119903 + 1|1 + 120582|sdot ln 10038161003816100381610038161003816100381610038161003816 119903 + |(1 minus 120582) (1 + 120582)|119903 minus |(1 minus 120582) (1 + 120582)| 10038161003816100381610038161003816100381610038161003816+ 1 +R1205821 + |120582|2 + 2R120582 (ln 119903210038161003816100381610038161003816|1 minus 120582|2 minus |1 + 120582|2 119903210038161003816100381610038161003816+ ln |1 minus 120582|2)

(30)

Therefore1003816100381610038161003816119891 (119911)1003816100381610038161003816 le 119903(|120582|2minus1)(1+|120582|2+2R120582)sdot 10038161003816100381610038161003816100381610038161003816 119903 + |(1 minus 120582) (1 + 120582)|119903 minus |(1 minus 120582) (1 + 120582)| 100381610038161003816100381610038161003816100381610038161|1+120582|sdot ( |1 minus 120582|2 119903210038161003816100381610038161003816|1 minus 120582|2 minus |1 + 120582|2 119903210038161003816100381610038161003816)

(1+R120582)(1+|120582|2+2R120582)

= 119903 10038161003816100381610038161003816100381610038161003816 119903 + |(1 minus 120582) (1 + 120582)|119903 minus |(1 minus 120582) (1 + 120582)| 100381610038161003816100381610038161003816100381610038161|1+120582|sdot ( 1100381610038161003816100381610038161 minus |(1 + 120582) (1 minus 120582)|2 119903210038161003816100381610038161003816)

(1+R120582)(1+|120582|2+2R120582)

= 119903 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)(1+|120582|2+2R120582)sdot 10038161003816100381610038161003816100381610038161003816119903 minus 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 1205821003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816minus1|1+120582|minus(1+R120582)(1+|120582|2+2R120582)sdot 10038161003816100381610038161003816100381610038161003816119903 + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161|1+120582|minus(1+R120582)(1+|120582|2+2R120582)= 119903 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2sdot 10038161003816100381610038161003816100381610038161003816119903 minus 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 1205821003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(minus|1+120582|minus1minusR120582)|1+120582|2sdot 10038161003816100381610038161003816100381610038161003816119903 + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 1205821003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(|1+120582|minus1minusR120582)|1+120582|2 = |119911|sdot 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2sdot 10038161003816100381610038161003816100381610038161003816|119911| minus 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 1205821003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(minus|1+120582|minus1minusR120582)|1+120582|2sdot (|119911| + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816)(|1+120582|minus1minusR120582)|1+120582|2

(31)

Obviously |(1 minus 120582)(1 + 120582)| ge 1 forR120582 le 0 Thus |119911| le |(1 minus120582)(1 + 120582)| So we obtain1003816100381610038161003816119891 (119911)1003816100381610038161003816 le |119911| 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2

sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 minus |119911|)(minus|1+120582|minus1minusR120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 + |119911|)(|1+120582|minus1minusR120582)|1+120582|2

(32)

Similarly by (19) we can get

1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge |119911| 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 minus |119911|)(|1+120582|minus1minusR120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 + |119911|)(minus|1+120582|minus1minusR120582)|1+120582|2

(33)

Theorem8 Let119891(119911) be an almost starlike function of complexorder 120582 on119863 with 120582 = minus1 Then

1 minus |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le 1 + |119911| |119911| 119890minus|119911| le 1003816100381610038161003816119891 (119911)1003816100381610038161003816 le |119911| 119890|119911|(1 minus |119911|) 119890minus|119911| le 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 le (1 + |119911|) 119890|119911|

(34)

Proof By (14) we have |2119892(119911) minus 1| le |119911| which follows|1199111198911015840(119911)119891(119911) minus 1| le |119911| Therefore

1 minus |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le 1 + |119911| 1 minus |119911| le R

1199111198911015840 (119911)119891 (119911) le 1 + |119911| (35)

For |119911| = 119903 we have1 minus 119903 le 119903 120597120597119903 ln 1003816100381610038161003816119891 (119911)1003816100381610038161003816 = R

1199111198911015840 (119911)119891 (119911) le 1 + 119903 (36)

Similar to Theorem 7 we get

int119903120576

119889119905119905 minus int119903120576119889119905 le int119903

120576

120597120597119905 ln 10038161003816100381610038161003816119891 (119905119890119894120579)10038161003816100381610038161003816 119889119905le int119903120576

119889119905119905 + int119903120576119889119905 (37)

6 Discrete Dynamics in Nature and Society

Let 120576 rarr 0 we obtain |119911|119890minus|119911| le |119891(119911)| le |119911|119890|119911| Hence (1 minus|119911|)119890minus|119911| le |1198911015840(119911)| le (1 + |119911|)119890|119911|Setting 120582 = 120572(120572 minus 1) in Theorems 7 and 8 we get the

following corollary

Corollary 9 Let 119891(119911) be an almost starlike function of order120572 Then |119911|(1 + (1 minus 2120572) |119911|)2(1minus120572)(1minus2120572) le 1003816100381610038161003816119891 (119911)1003816100381610038161003816le |119911|(1 minus (1 minus 2120572) |119911|)2(1minus120572)(1minus2120572) 120572 isin [0 1) 12

|119911| 119890minus|119911| le 1003816100381610038161003816119891 (119911)1003816100381610038161003816 le |119911| 119890|119911| 120572 = 12 1 minus |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le 1 + |119911| (1 minus |119911|) 119890minus|119911| le 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 le (1 + |119911|) 119890|119911|

120572 = 12

(38)

Theorem 10 Let 119891(119911) be an almost starlike function ofcomplex order 120582 on119863 with 120582 isin C minus1 Re 120582 le 0 Then1 minus |119911|1 + |(1 + 120582) (1 minus 120582)| |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816le 1 + |119911|1 minus |(1 + 120582) (1 minus 120582)| |119911|

(1 minus |119911|) 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2+1sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 minus |119911|)(|1+120582|minus1minusR120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 + |119911|)(minus|1+120582|minus1minusR120582)|1+120582|2minus1 le 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816le (1 + |119911|) 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2+1sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 + |119911|)(|1+120582|minus1minusR120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 minus |119911|)(minus|1+120582|minus1minusR120582)|1+120582|2minus1

(39)

Proof By (17) we have100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) minus 1100381610038161003816100381610038161003816100381610038161003816 le 10038161003816100381610038161003816100381610038161003816 1 + 1205821 minus 12058210038161003816100381610038161003816100381610038161003816 |119911| 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) + 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038161003816 (40)

Since 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 minus 1 le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) minus 1100381610038161003816100381610038161003816100381610038161003816 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) + 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038161003816 le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 (41)

then we have100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 minus 1 le (100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816) 10038161003816100381610038161003816100381610038161003816 1 + 1205821 minus 12058210038161003816100381610038161003816100381610038161003816 |119911| (42)

which follows100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le 1 + |119911|1 minus |(1 + 120582) (1 minus 120582)| |119911| (43)

On the other hand we also have

1 minus 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le (100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816) 10038161003816100381610038161003816100381610038161003816 1 + 1205821 minus 12058210038161003816100381610038161003816100381610038161003816 |119911| (44)

which leads to100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 ge 1 minus |119911|1 + |(1 + 120582) (1 minus 120582)| |119911| (45)

Hence 1 minus |119911|1 + |(1 + 120582) (1 minus 120582)| |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816le 1 + |119911|1 minus |(1 + 120582) (1 minus 120582)| |119911| (46)

Then by Theorem 7 we obtain the desired conclusion (39)

Setting 120582 = 120572(120572 minus 1) inTheorem 10 we get the followingresult

Corollary 11 Let 119891(119911) be an almost starlike function of order120572 with 120572 isin [0 1) 12 Then1 minus |119911|1 + |1 minus 2120572| |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le 1 + |119911|1 minus |1 minus 2120572| |119911| 1 minus |119911|1 + |1 minus 2120572| |119911| |119911|(1 + (1 minus 2120572) |119911|)2(1minus120572)(1minus2120572)le 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816le 1 + |119911|1 minus |1 minus 2120572| |119911| |119911|(1 minus (1 minus 2120572) |119911|)2(1minus120572)(1minus2120572) (47)

3 Coefficient Estimates of Almost StarlikeFunctions of Complex Order 120582

Lemma 12 (see [21]) Let 119901(119911) = 1 + suminfin119899=1 119888119899119911119899 be aholomorphic function on 119863 with R119901(119911) gt 0 Then |119888119899| le2 (119899 = 1 2 )

Discrete Dynamics in Nature and Society 7

Lemma 13 (see [22]) Let119901(119911) = 1+suminfin119899=1 119888119899119911119899 be holomorphicon119863 ThenR119901(119911) gt 0 if and only if

119901 (119911) = 120585 minus 1120585 + 1 120585 isin 120597119863 (48)

Theorem 14 Let 119891(119911) = 119911 + suminfin119899=2 119886119899119911119899 be an almost starlikefunction of complex order 120582 with 120582 isin C Re 120582 le 0 on119863 Then10038161003816100381610038161198861198991003816100381610038161003816 le 119899minus1prod

119895=2

( 2|1 minus 120582| + 119895 minus 2119895 minus 1) 119899 = 2 3 (49)

Proof Since 119891(119911) is an almost starlike function of complexorder 120582 then

R [(1 minus 120582) 119891 (119911)1199111198911015840 (119911) + 120582] gt 0 119911 isin 119863 (50)

Let 119901(119911) = (1 minus 120582)(119891(119911)1199111198911015840(119911)) + 120582 Then we have1199111198911015840 (119911) (119901 (119911) minus 120582) = (1 minus 120582) 119891 (119911) (51)

and 119901(0) = 1 andR119901(119911) gt 0 Let 119901(119911) = 1 + suminfin119899=1 119888119899119911119899 Then10038161003816100381610038161198881198991003816100381610038161003816 le 2 (119899 = 1 2 ) (52)

by Lemma 12 For 119891(119911) = 119911 + suminfin119899=2 119886119899119911119899 (51) follows119911(1 + infinsum

119899=2

119899119886119899119911119899minus1)(1 minus 120582 + infinsum119899=1

119888119899119911119899)= (1 minus 120582)(119911 + infinsum

119899=2

119886119899119911119899) (53)

Therefore(1 minus 120582) 119911 + [1198881 + 21198862 (1 minus 120582)] 1199112 + sdot sdot sdot + [119888119899minus1+ 21198862119888119899minus2 + sdot sdot sdot + (119899 minus 1) 119886119899minus11198881 + 119899119886119899 (1 minus 120582)] 119911119899+ sdot sdot sdot = (1 minus 120582) 119911 + 1198862 (1 minus 120582) 1199112 + sdot sdot sdot + 119886119899 (1 minus 120582)sdot 119911119899 + sdot sdot sdot (54)

So 119886119899 = 119888119899minus1 + 21198862119888119899minus2 + sdot sdot sdot + (119899 minus 1) 119886119899minus11198881(119899 minus 1) (120582 minus 1) (55)

By (52) we get10038161003816100381610038161198861198991003816100381610038161003816 le 2 [1 + 2 100381610038161003816100381611988621003816100381610038161003816 + sdot sdot sdot + (119899 minus 1) 1003816100381610038161003816119886119899minus11003816100381610038161003816](119899 minus 1) |1 minus 120582| (56)

Bymathematical induction we obtain the desired conclusion

Theorem 15 Let 119891(119911) be an almost starlike function ofcomplex order 120582 with 120582 isin C Re 120582 le 0 on 119863 and let 119911 = 0be the zero of order 119896 + 1 (119896 isin N) of 119891(119911) minus 119911 Let 119891(119911) =119911 + suminfin119899=119896+1 119886119899119911119899 Then1003816100381610038161003816119886119899+11003816100381610038161003816 le 2|120582 + 119899| (119899 = 119896 119896 + 1 ) (57)

Proof Since 119891(119911) = 119911 + suminfin119899=119896+1 119886119899119911119899 similar to Theorem 14by (51) we obtain

119911(1 + infinsum119899=119896+1

119899119886119899119911119899minus1)(1 minus 120582 + infinsum119899=1

119888119899119911119899)= (1 minus 120582)(119911 + infinsum

119899=119896+1

119886119899119911119899) (58)

which follows(1 minus 120582) 119911 + 11988811199112 + sdot sdot sdot + 119888119896minus1119911119896+ [119888119896 + (119896 + 1) 119886119896+1] 119911119896+1 + sdot sdot sdot+ [119888119899 + (119899 + 1) 119886119899+1] 119911119899+1 + sdot sdot sdot= (1 minus 120582) 119911 + (1 minus 120582) 119886119896+1119911119896+1 + sdot sdot sdot+ (1 minus 120582) 119886119899+1119911119899+1 + sdot sdot sdot (59)

Therefore 1198881 = sdot sdot sdot = 119888119896minus1 = 0119888119899 = (minus120582 minus 119899) 119886119899+1 (119899 = 119896 119896 + 1 ) (60)

By (52) we get the desired conclusion

Theorem 16 119891(119911) = 119911 + suminfin119899=2 119886119899119911119899 is an almost starlikefunction of complex order 120582 with 120582 isin C and Re 120582 le 0 on 119863if and only ifinfinsum119899=2

[(1 + (119899 minus 1) 120582) (120585 + 1) + 119899 (1 minus 120585)] 119886119899119911119899 = minus2(120585 isin 120597119863 119899 = 2 3 ) (61)

Proof Let 119901(119911) = (1 minus 120582)(119891(119911)1199111198911015840(119911)) + 120582 Since 119891(119911) isan almost starlike function of complex order 120582 then 119901(119911)is holomorphic on 119863 and 119901(0) = 0 and R119901(119911) gt 0 ByLemma 13 we get(1 minus 120582) 119891 (119911)1199111198911015840 (119911) + 120582 = 120585 minus 1120585 + 1 120585 isin 120597119863 (62)

which follows(1 minus 120582) (120585 + 1) 119891 (119911) + 120582 (120585 + 1) 1199111198911015840 (119911)= (120585 minus 1) 1199111198911015840 (119911) (63)

For 119891(119911) = 119911 + suminfin119899=2 119886119899119911119899 we have(1 minus 120582) (120585 + 1) (119911 + infinsum119899=2

119886119899119911119899)+ 120582 (120585 + 1) (119911 + infinsum

119899=2

119899119886119899119911119899)= (120585 minus 1) (119911 + infinsum

119899=2

119899119886119899119911119899) (64)

which leads to the desired conclusion

8 Discrete Dynamics in Nature and Society

Remark 17 Setting 120582 = 120572(120572 minus 1) in Theorems 14ndash16 weget the corresponding results for almost starlike functions oforder 1205724 The Invariance of Almost Starlike Mappings

of Complex Order 120582In this section we mainly discuss the invariance of almoststarlike mappings of complex order 120582 on Reinhardt domainsand on the unit ball in complex Banach spaces under somegeneralized Roper-Suffridge operators

Lemma 18 (see [23]) Let Ω sub C119899 be a bounded andcompletely Reinhardt domain whose Minkowski functional120588(119911) is 1198621 except for a lower-dimensional manifold Ω0 Thenfor forall119911 = (1199111 119911119899) isin Ω Ω0 we have120597120588 (119911)120597119911 = (120597120588 (119911)1205971199111 120597120588 (119911)1205971199112 120597120588 (119911)120597119911119899 )

120597120588 (119911)120597119911119895 119911119895 ge 0120588 (119911) = 2 119899sum

119895=1

120597120588 (119911)120597119911119895 119911119895(65)

Theorem 19 Let 119891(119911) be an almost starlike function of com-plex order 120582 on 119863 with 120582 isin C and Re 120582 le 0 Let Ω sub C119899 be abounded and completely Reinhardt domain whose Minkowskifunctional 120588(119911) is 1198621 on Ω 0 Let

119865 (119911) = (119903119891(1199111119903 ) (119903119891 (1199111119903)1199111 )1205732sdot 1199112 (119903119891 (1199111119903)1199111 )120573119899 119911119899)1015840

(66)

where 120573119895 isin [0 1] 119903 = sup|1199111| 119911 = (1199111 119911119899)1015840 isin Ω and(119903119891(1199111119903)1199111)120573119895 |1199111=0 = 1 (119895 = 2 119899) Then 119865(119911) is an almoststarlike mapping of complex order 120582 on ΩProof For

119865 (119911) = (119903119891(1199111119903 ) (119903119891 (1199111119903)1199111 )1205732sdot 1199112 (119903119891 (1199111119903)1199111 )120573119899 119911119899)1015840

(67)

immediately we have

119869119865 (119911) =(((((((((((

1198911015840 (1199111119903 ) 0 sdot sdot sdot 01205732119903 [1198911015840 (1199111119903)119891 (1199111119903) minus 1199031199111 ](119903119891 (1199111119903)1199111 )1205732 1199112 (119903119891 (1199111119903)1199111 )1205732 sdot sdot sdot 0

sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot120573119899119903 [1198911015840 (1199111119903)119891 (1199111119903) minus 1199031199111](119903119891 (1199111119903)1199111 )120573119899 119911119899 0 sdot sdot sdot (119903119891 (1199111119903)1199111 )120573119899

)))))))))))

(68)

Therefore

119869minus1119865 (119911) =(((((((((((

11198911015840 (1199111119903) 0 sdot sdot sdot 01205732119903 [ 11990311991111198911015840 (1199111119903) minus 1119891 (1199111119903)] 1199112 (119903119891 (1199111119903)1199111 )minus1205732 sdot sdot sdot 0

sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot120573119899119903 [ 11990311991111198911015840 (1199111119903) minus 1119891 (1199111119903)] 119911119899 0 sdot sdot sdot (119903119891 (1199111119903)1199111 )minus120573119899

)))))))))))

(69)

Discrete Dynamics in Nature and Society 9

Then 119869minus1119865 (119911) 119865 (119911)= (119903119891 (1199111119903)1198911015840 (1199111119903) (1 minus 1205732 + 1205732 119903119891 (1199111119903)11991111198911015840 (1199111119903))sdot 1199112 (1 minus 120573119899 + 120573119899 119903119891 (1199111119903)11991111198911015840 (1199111119903)) 119911119899)1015840

(70)

By Lemma 18 we get(1 minus 120582) 2120597120588120597119911 (119911) 119869minus1119865 (119911) 119865 (119911) = 2 (1 minus 120582)sdot [[ 119903119891 (1199111119903)11991111198911015840 (1199111119903) 120597120588 (119911)1205971199111 1199111+ 119899sum119895=2

(1 minus 120573119895 + 120573119895 119903119891 (1199111119903)11991111198911015840 (1199111119903)) 120597120588 (119911)120597119911119895 119911119895]] (71)

Since119891(119911) is an almost starlike function of complex order120582 thenR[(1 minus 120582) 119903119891 (1199111119903)11991111198911015840 (1199111119903)] ge minusR120582 (72)

Therefore by Lemma 18 we obtain

R [(1 minus 120582) 2120597120588120597119911 (119911) 119869minus1119865 (119911) 119865 (119911)] = R[(1 minus 120582)sdot 119903119891 (1199111119903)11991111198911015840 (1199111119903)] 2120597120588 (119911)1205971199111 1199111 + 119899sum

119895=2

(1 minus 120573119895)R (1 minus 120582)+ 120573119895R[(1 minus 120582) 119903119891 (1199111119903)11991111198911015840 (1199111119903)] 2120597120588 (119911)120597119911119895 119911119895 ge minusR120582sdot 2120597120588 (119911)1205971199111 1199111 + 119899sum

119895=2

(1 minus 120573119895) (1 minusR120582) + 120573119895 (minusR120582)sdot 2120597120588 (119911)120597119911119895 119911119895 = (minusR120582) [2120597120588 (119911)1205971199111 1199111 + 2120597120588 (119911)120597119911119895 119911119895]+ 119899sum119895=2

(1 minus 120573119895) 2120597120588 (119911)120597119911119895 119911119895 ge (minusR120582) 120588 (119911)

(73)

Hence 119865(119911) is an almost starlike mapping of complex order 120582onΩ by Definition 4

Theorem 20 Let 119891119895 (119895 = 1 2 119899) be almost starlikefunctions of complex order 120582 on 119863 with 120582 isin C and Re 120582 lt 0Let Ω sub C119899 be a bounded starlike and circular domain whoseMinkowski functional 120588(119911) is1198621 onΩ0 Let 119903119895 be the radiusof the disk 119880119895 = 119911119895 isin C 119911 = (1199111 119911119895 119911119899)1015840 isin Ω whosecenter is zero Let

119865 (119911) = 119911 119899prod119895=1

(119903119895119891119895 (119911119895119903119895)119911119895 )120582119894 (74)

where 120582119894119895 ge 0 sum119899119895=1 120582119894119895 = 1 and (119891119895(119911119895)119911119895)120582119894119895 |119911119895=0 = 1 (119894 119895 =1 2 119899)Then119865(119911) is an almost starlikemapping of complexorder 120582 on ΩProof Let

120577119895 = 119911119895119903119895 119892 (119911) = 119899prod

119895=1

(119891119895 (120577119895)120577119895 )120582119894 (75)

which follows 119865(119911) = 119911119892(119911) Then 119869119865(119911) = 119869119892(119911)119911 + 119892(119911) Bydirect computation we get

119869119892 (119911) 119911 = 119892 (119911) [[119899sum119895=1

1205821198951198911015840119895 (119911119895119903119895) 119911119895119903119895119891119895 (119911119895119903119895) minus 1]]= 119892 (119911) [[

119899sum119895=1

1205821198951198911015840119895 (120577119895) 120577119895119891119895 (120577119895) minus 1]] (76)

Since 119891119895 are almost starlike functions of complex order 120582 on119863 then

R[[(1 minus 120582) 119891119895 (120577119895)1205771198951198911015840119895 (120577119895)]] ge minusR120582 gt 0 (77)

Therefore

R[ 11 minus 120582 (1 + 119869119892 (119911) 119911119892 (119911) )]= R[[ 11 minus 120582 119899sum

119895=1

1205821198951198911015840119895 (120577119895) 120577119895119891119895 (120577119895) ]]= 119899sum119895=1

120582119895R[[ 11 minus 120582 1198911015840119895 (120577119895) 120577119895119891119895 (120577119895) ]] gt 0(78)

Then 119869119892(119911)119911+119892(119911) = 0 It is obvious that119865(0) = 0 and 119869119865(0) =119868 so 119865(119911) is a normalized locally biholomorphic mapping onΩ and

119869minus1119865 (119911) 119865 (119911) = 119892 (119911) 119911119869119892 (119911) 119911 + 119892 (119911) (79)

Let

119902119895 (120577119895) = 2Re (minus120582) 1198911015840119895 (120577119895) 120577119895(1 minus 120582) 119891119895 (120577119895) minus 1 (80)

(77) follows 100381610038161003816100381610038161003816100381610038161003816100381610038162Re (minus120582) 1198911015840119895 (120577119895) 120577119895(1 minus 120582) 119891119895 (120577119895) minus 110038161003816100381610038161003816100381610038161003816100381610038161003816 le 1 (81)

10 Discrete Dynamics in Nature and Society

that is |119902119895(120577119895)| le 1 For Re 120582 lt 0 by (76) (79) and Lemma 18we obtain1003816100381610038161003816100381610038161003816100381610038162R (minus120582) 1(1 minus 120582) (2120588 (119911)) (120597120588120597119911) (119911) 119869minus1119865 (119911) 119865 (119911)

minus 1100381610038161003816100381610038161003816100381610038161003816 =100381610038161003816100381610038161003816100381610038161003816100381610038162R (minus120582) 11 minus 120582 119899sum

119895=1

1205821198951198911015840119895 (120577119895) 120577119895119891119895 (120577119895) minus 110038161003816100381610038161003816100381610038161003816100381610038161003816= 10038161003816100381610038161003816100381610038161003816100381610038161003816 119899sum119895=1120582119895119902119895 (120577119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816 le 119899sum119895=1120582119895 = 1(82)

which leads to

R [(1 minus 120582) 2120597120588120597119911 (119911) 119869minus1119865 (119911) 119865 (119911)] ge minusR120582120588 (119911) (83)

Hence 119865(119911) is an almost starlike mapping of complex order 120582onΩ by Definition 4

Similar to Theorem 20 we can get the following conclu-sion

Theorem 21 Let 119891119895 (119895 = 1 2 119899) be almost starlikefunctions of complex order 120582 on 119863 with 120582 isin C and Re 120582 lt 0Let

119865 (119909) = 119909 119899prod119895=1

(119891119895 (119879119906119895 (119909))119879119906119895 (119909) )120582119895 119909 isin 119861 (84)

where 120582119894119895 ge 0 sum119899119895=1 120582119894119895 = 1 (119891119895(119909119895)119909119895)120582119894119895 |119909119895=0 = 1 (119909119895 =119879119906119895(119909) 119894 119895 = 1 2 119899) and 119906119895 is the unit vector of thecomplex Banach space 119883 Then 119865(119909) is an almost starlikemapping of complex order 120582 on the unit ball 119861 in complexBanach space119883

Theorem 22 Let 119891 be an almost starlike function of complexorder 120582 on119863 with 120582 isin C and Re 120582 le 0 Let

119865 (119909) = 119899minus1sum119895=1

(119891(1198791199091 (119909))1198791199091 (119909) )120573119895 119879119909119895 (119909) 119909119895 + 119909minus 119899minus1sum119895=1

119879119909119895 (119909) 119909119895 119909 isin 119861 (85)

where 1199091 isin 119861 1199091 = 1 120573119895 isin [0 1] (119891(119911)119911)120573119895 |119911=0 = 1 (119895 =1 2 119899 minus 1) and 1199091 119909119899 are linearly independent and119879119909119894 isin 119883lowast such that 119879119909119894 = 1 119879119909119894(119909119894) = 1 and 119879119909119894(119909119895) =0 (119894 = 119895) for forall119909119894 Then 119865(119909) is an almost starlike mapping ofcomplex order 120582 on the unit ball 119861 in complex Banach space119883

Proof From [24] we get 119865(119909) is a normalized biholomorphicmapping on 119861 and119909 119879119909 [119863119865 (119909)minus1 119865 (119909)]= 1199092

+ 119899minus1sum119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162 [ 119891 (1198791199091 (119909))1198791199091 (119909) 1198911015840 (1198791199091 (119909)) minus 1] (86)

Since119891 is an almost starlike function of complex order120582 thenR[[(1 minus 120582) 119891 (119879119909119895 (119909))119879119909119895 (119909) 1198911015840 (119879119909119895 (119909))]] ge minusR120582 (87)

Therefore

R (1 minus 120582) 119879119909 [119863119865 (119909)minus1 119865 (119909)] = (1 minusR120582) 119909+ 119899minus1sum119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162119909sdotR(1 minus 120582) [ 119891 (1198791199091 (119909))1198791199091 (119909) 1198911015840 (1198791199091 (119909)) minus 1]ge (1 minusR120582) 119909 + 119899minus1sum

119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162119909sdot [R (minus120582) minus (1 minusR120582)] = minusR120582 119909+ 1119909 [[1199092 minus

119899minus1sum119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162]] ge minusR120582 119909

(88)

which lead to the desired conclusion by Definition 5

Similar to Theorem 22 we can get the following conclu-sion

Theorem 23 Let 119891 be an almost starlike function of complexorder 120582 on119863 with 120582 isin C Re 120582 le 0 Let119865 (119909) = Φ120573 (119891) (119909)= 119891 (1198791199091 (119909)) 1199091

+ (119891 (1198791199091 (119909))1198791199091 (119909) )120573 (119909 minus 1198791199091 (119909) 1199091) (89)

where 0 le 120573 le 1 1199091 isin 119861 1199091 = 1 (119891(1199111)1199111)120573|1199111=0 = 1 and119891(1199111) = 0 for 1199111 = 0 Then 119865(119909) is an almost starlike mappingof complex order 120582 on 119861Remark 24 Setting 120582 = 120572(120572 minus 1) in Theorems 19ndash23 weget the corresponding results with respect to almost starlikemappings of order 120572

Discrete Dynamics in Nature and Society 11

Competing Interests

The authors declare that they have no conflict of interests

Acknowledgments

This work is supported by NSF of China (no U1204618) andScience and Technology Research Projects of Henan Provin-cial Education Department (nos 14B110015 and 14B110016)

References

[1] C Pommerenke Univalent Functions Vandenhoeck amp Ru-precht Gottingen Germany 1975

[2] H Cartan ldquoSur la possibilite drsquoentendre aux functions de plus-ieurs variables complexes la theorie des functions univalentsrdquoin Lecons sur les Functions Univalents onMutivalents P MontelEd Gauthier-Villar Paris France 1933

[3] R W Barnard C H FitzGerald and S Gong ldquoA distortionthoerem for biholomorphicmappings inC2rdquoTransactions of theAmerican Mathematical Society vol 344 pp 902ndash924 1994

[4] T S Liu andW J Zhang ldquoA distortion thoerem for biholomor-phic convex mappings in C119899rdquo Chinese Journal of ContemporaryMathematics vol 20 no 3 pp 421ndash430 1999

[5] S Gong S K Wang and Q H Yu ldquoBiholomorphic convexmappings of ball in C119899rdquo Pacific Journal of Mathematics vol 161no 2 pp 287ndash306 1993

[6] S Gong and T S Liu ldquoDistortion theorems for biholomorphicconvex mappings on bounded convex circular domainsrdquo Chi-nese Annals of Mathematics Series B vol 20 no 3 pp 297ndash3041999

[7] I Graham and D Varolin ldquoBloch constants in one and severalvariablesrdquoPacific Journal ofMathematics vol 174 no 2 pp 347ndash357 1996

[8] P L DurenUnivalent Functions Springer New York NY USA1983

[9] T H Gronwall ldquoSome remarks on conformal representationrdquoAnnals of Mathematics vol 16 no 1ndash4 pp 72ndash76 191415

[10] M I Robertson ldquoOn the theory of univalent functionsrdquo Annalsof Mathematics vol 37 no 2 pp 374ndash408 1936

[11] A V Boyd ldquoCoefficient estimates for starlike functions of order120572rdquo Proceedings of the AmericanMathematical Society vol 17 pp1016ndash1018 1966

[12] H Silverman ldquoUnivalent functions with negative coefficientsrdquoProceedings of the American Mathematical Society vol 51 pp109ndash116 1975

[13] H M Srivastava and S Owa ldquoCertain subclasses of starlikefunctions Irdquo Journal ofMathematical Analysis and Applicationsvol 161 no 2 pp 405ndash415 1991

[14] K A Roper and T J Suffridge ldquoConvex mappings on the unitball ofC119899rdquo Journal drsquoAnalyseMathematique vol 65 pp 333ndash3471995

[15] I Graham and G Kohr ldquoUnivalent mappings associated withthe Roper-Suffridge extension operatorrdquo Journal drsquoAnalyseMathematique vol 81 pp 331ndash342 2000

[16] S Gong and T S Liu ldquoThe generalized Roper-Suffridge exten-sion operatorrdquo Journal of Mathematical Analysis and Applica-tions vol 284 no 2 pp 425ndash434 2003

[17] G Kohr ldquoLoewner chains and a modification of the Roper-Suffridge extension operatorrdquo Mathematica vol 71 no 1 pp41ndash48 2006

[18] J R Muir ldquoA class of Loewner chain preserving extensionoperatorsrdquo Journal of Mathematical Analysis and Applicationsvol 337 no 2 pp 862ndash879 2008

[19] Y H Zhao Almost Starlike Mappings of Complex Order 120582 onthe Unit Ball of a Complex Banach Space Zhejiang NormalUniversity Jinhua China 2013

[20] L V Ahlfors Complex Analysis McGraw-Hill New York NYUSA 3rd edition 1978

[21] L Bieberbach ldquoUber einige extremal problem in gebiete derkonformen abbildungrdquo Mathematische Annalen vol 77 no 2pp 153ndash172 1916

[22] T Hayami S Owa and H M Srivastava ldquoCoefficient inequal-ities for certain classes of analytic and univalent functionsrdquoJournal of Inequalities in Pure and Applied Mathematics vol 8no 4 pp 1ndash10 2007

[23] M S Liu and Y C Zhu ldquoGeneralized Roper-Suffridge oper-ators on bounded and complete Reinhardt domainsrdquo Sciencein China vol 37 no 10 pp 1193ndash1206 2007

[24] Y Y Cui and C J Wang ldquoThe generalized Roper-Suffridgeextension operators in complex Banach spacerdquo Journal ofMathematics vol 34 no 3 pp 515ndash520 2014

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Stochastic AnalysisInternational Journal of

6 Discrete Dynamics in Nature and Society

Let 120576 rarr 0 we obtain |119911|119890minus|119911| le |119891(119911)| le |119911|119890|119911| Hence (1 minus|119911|)119890minus|119911| le |1198911015840(119911)| le (1 + |119911|)119890|119911|Setting 120582 = 120572(120572 minus 1) in Theorems 7 and 8 we get the

following corollary

Corollary 9 Let 119891(119911) be an almost starlike function of order120572 Then |119911|(1 + (1 minus 2120572) |119911|)2(1minus120572)(1minus2120572) le 1003816100381610038161003816119891 (119911)1003816100381610038161003816le |119911|(1 minus (1 minus 2120572) |119911|)2(1minus120572)(1minus2120572) 120572 isin [0 1) 12

|119911| 119890minus|119911| le 1003816100381610038161003816119891 (119911)1003816100381610038161003816 le |119911| 119890|119911| 120572 = 12 1 minus |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le 1 + |119911| (1 minus |119911|) 119890minus|119911| le 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 le (1 + |119911|) 119890|119911|

120572 = 12

(38)

Theorem 10 Let 119891(119911) be an almost starlike function ofcomplex order 120582 on119863 with 120582 isin C minus1 Re 120582 le 0 Then1 minus |119911|1 + |(1 + 120582) (1 minus 120582)| |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816le 1 + |119911|1 minus |(1 + 120582) (1 minus 120582)| |119911|

(1 minus |119911|) 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2+1sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 minus |119911|)(|1+120582|minus1minusR120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 + |119911|)(minus|1+120582|minus1minusR120582)|1+120582|2minus1 le 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816le (1 + |119911|) 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038162(1+R120582)|1+120582|2+1sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 + |119911|)(|1+120582|minus1minusR120582)|1+120582|2sdot (10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 minus |119911|)(minus|1+120582|minus1minusR120582)|1+120582|2minus1

(39)

Proof By (17) we have100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) minus 1100381610038161003816100381610038161003816100381610038161003816 le 10038161003816100381610038161003816100381610038161003816 1 + 1205821 minus 12058210038161003816100381610038161003816100381610038161003816 |119911| 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) + 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038161003816 (40)

Since 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 minus 1 le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) minus 1100381610038161003816100381610038161003816100381610038161003816 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) + 1 minus 1205821 + 120582100381610038161003816100381610038161003816100381610038161003816 le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816 (41)

then we have100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 minus 1 le (100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816) 10038161003816100381610038161003816100381610038161003816 1 + 1205821 minus 12058210038161003816100381610038161003816100381610038161003816 |119911| (42)

which follows100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le 1 + |119911|1 minus |(1 + 120582) (1 minus 120582)| |119911| (43)

On the other hand we also have

1 minus 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le (100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 + 10038161003816100381610038161003816100381610038161003816 1 minus 1205821 + 12058210038161003816100381610038161003816100381610038161003816) 10038161003816100381610038161003816100381610038161003816 1 + 1205821 minus 12058210038161003816100381610038161003816100381610038161003816 |119911| (44)

which leads to100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 ge 1 minus |119911|1 + |(1 + 120582) (1 minus 120582)| |119911| (45)

Hence 1 minus |119911|1 + |(1 + 120582) (1 minus 120582)| |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816le 1 + |119911|1 minus |(1 + 120582) (1 minus 120582)| |119911| (46)

Then by Theorem 7 we obtain the desired conclusion (39)

Setting 120582 = 120572(120572 minus 1) inTheorem 10 we get the followingresult

Corollary 11 Let 119891(119911) be an almost starlike function of order120572 with 120572 isin [0 1) 12 Then1 minus |119911|1 + |1 minus 2120572| |119911| le 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) 100381610038161003816100381610038161003816100381610038161003816 le 1 + |119911|1 minus |1 minus 2120572| |119911| 1 minus |119911|1 + |1 minus 2120572| |119911| |119911|(1 + (1 minus 2120572) |119911|)2(1minus120572)(1minus2120572)le 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816le 1 + |119911|1 minus |1 minus 2120572| |119911| |119911|(1 minus (1 minus 2120572) |119911|)2(1minus120572)(1minus2120572) (47)

3 Coefficient Estimates of Almost StarlikeFunctions of Complex Order 120582

Lemma 12 (see [21]) Let 119901(119911) = 1 + suminfin119899=1 119888119899119911119899 be aholomorphic function on 119863 with R119901(119911) gt 0 Then |119888119899| le2 (119899 = 1 2 )

Discrete Dynamics in Nature and Society 7

Lemma 13 (see [22]) Let119901(119911) = 1+suminfin119899=1 119888119899119911119899 be holomorphicon119863 ThenR119901(119911) gt 0 if and only if

119901 (119911) = 120585 minus 1120585 + 1 120585 isin 120597119863 (48)

Theorem 14 Let 119891(119911) = 119911 + suminfin119899=2 119886119899119911119899 be an almost starlikefunction of complex order 120582 with 120582 isin C Re 120582 le 0 on119863 Then10038161003816100381610038161198861198991003816100381610038161003816 le 119899minus1prod

119895=2

( 2|1 minus 120582| + 119895 minus 2119895 minus 1) 119899 = 2 3 (49)

Proof Since 119891(119911) is an almost starlike function of complexorder 120582 then

R [(1 minus 120582) 119891 (119911)1199111198911015840 (119911) + 120582] gt 0 119911 isin 119863 (50)

Let 119901(119911) = (1 minus 120582)(119891(119911)1199111198911015840(119911)) + 120582 Then we have1199111198911015840 (119911) (119901 (119911) minus 120582) = (1 minus 120582) 119891 (119911) (51)

and 119901(0) = 1 andR119901(119911) gt 0 Let 119901(119911) = 1 + suminfin119899=1 119888119899119911119899 Then10038161003816100381610038161198881198991003816100381610038161003816 le 2 (119899 = 1 2 ) (52)

by Lemma 12 For 119891(119911) = 119911 + suminfin119899=2 119886119899119911119899 (51) follows119911(1 + infinsum

119899=2

119899119886119899119911119899minus1)(1 minus 120582 + infinsum119899=1

119888119899119911119899)= (1 minus 120582)(119911 + infinsum

119899=2

119886119899119911119899) (53)

Therefore(1 minus 120582) 119911 + [1198881 + 21198862 (1 minus 120582)] 1199112 + sdot sdot sdot + [119888119899minus1+ 21198862119888119899minus2 + sdot sdot sdot + (119899 minus 1) 119886119899minus11198881 + 119899119886119899 (1 minus 120582)] 119911119899+ sdot sdot sdot = (1 minus 120582) 119911 + 1198862 (1 minus 120582) 1199112 + sdot sdot sdot + 119886119899 (1 minus 120582)sdot 119911119899 + sdot sdot sdot (54)

So 119886119899 = 119888119899minus1 + 21198862119888119899minus2 + sdot sdot sdot + (119899 minus 1) 119886119899minus11198881(119899 minus 1) (120582 minus 1) (55)

By (52) we get10038161003816100381610038161198861198991003816100381610038161003816 le 2 [1 + 2 100381610038161003816100381611988621003816100381610038161003816 + sdot sdot sdot + (119899 minus 1) 1003816100381610038161003816119886119899minus11003816100381610038161003816](119899 minus 1) |1 minus 120582| (56)

Bymathematical induction we obtain the desired conclusion

Theorem 15 Let 119891(119911) be an almost starlike function ofcomplex order 120582 with 120582 isin C Re 120582 le 0 on 119863 and let 119911 = 0be the zero of order 119896 + 1 (119896 isin N) of 119891(119911) minus 119911 Let 119891(119911) =119911 + suminfin119899=119896+1 119886119899119911119899 Then1003816100381610038161003816119886119899+11003816100381610038161003816 le 2|120582 + 119899| (119899 = 119896 119896 + 1 ) (57)

Proof Since 119891(119911) = 119911 + suminfin119899=119896+1 119886119899119911119899 similar to Theorem 14by (51) we obtain

119911(1 + infinsum119899=119896+1

119899119886119899119911119899minus1)(1 minus 120582 + infinsum119899=1

119888119899119911119899)= (1 minus 120582)(119911 + infinsum

119899=119896+1

119886119899119911119899) (58)

which follows(1 minus 120582) 119911 + 11988811199112 + sdot sdot sdot + 119888119896minus1119911119896+ [119888119896 + (119896 + 1) 119886119896+1] 119911119896+1 + sdot sdot sdot+ [119888119899 + (119899 + 1) 119886119899+1] 119911119899+1 + sdot sdot sdot= (1 minus 120582) 119911 + (1 minus 120582) 119886119896+1119911119896+1 + sdot sdot sdot+ (1 minus 120582) 119886119899+1119911119899+1 + sdot sdot sdot (59)

Therefore 1198881 = sdot sdot sdot = 119888119896minus1 = 0119888119899 = (minus120582 minus 119899) 119886119899+1 (119899 = 119896 119896 + 1 ) (60)

By (52) we get the desired conclusion

Theorem 16 119891(119911) = 119911 + suminfin119899=2 119886119899119911119899 is an almost starlikefunction of complex order 120582 with 120582 isin C and Re 120582 le 0 on 119863if and only ifinfinsum119899=2

[(1 + (119899 minus 1) 120582) (120585 + 1) + 119899 (1 minus 120585)] 119886119899119911119899 = minus2(120585 isin 120597119863 119899 = 2 3 ) (61)

Proof Let 119901(119911) = (1 minus 120582)(119891(119911)1199111198911015840(119911)) + 120582 Since 119891(119911) isan almost starlike function of complex order 120582 then 119901(119911)is holomorphic on 119863 and 119901(0) = 0 and R119901(119911) gt 0 ByLemma 13 we get(1 minus 120582) 119891 (119911)1199111198911015840 (119911) + 120582 = 120585 minus 1120585 + 1 120585 isin 120597119863 (62)

which follows(1 minus 120582) (120585 + 1) 119891 (119911) + 120582 (120585 + 1) 1199111198911015840 (119911)= (120585 minus 1) 1199111198911015840 (119911) (63)

For 119891(119911) = 119911 + suminfin119899=2 119886119899119911119899 we have(1 minus 120582) (120585 + 1) (119911 + infinsum119899=2

119886119899119911119899)+ 120582 (120585 + 1) (119911 + infinsum

119899=2

119899119886119899119911119899)= (120585 minus 1) (119911 + infinsum

119899=2

119899119886119899119911119899) (64)

which leads to the desired conclusion

8 Discrete Dynamics in Nature and Society

Remark 17 Setting 120582 = 120572(120572 minus 1) in Theorems 14ndash16 weget the corresponding results for almost starlike functions oforder 1205724 The Invariance of Almost Starlike Mappings

of Complex Order 120582In this section we mainly discuss the invariance of almoststarlike mappings of complex order 120582 on Reinhardt domainsand on the unit ball in complex Banach spaces under somegeneralized Roper-Suffridge operators

Lemma 18 (see [23]) Let Ω sub C119899 be a bounded andcompletely Reinhardt domain whose Minkowski functional120588(119911) is 1198621 except for a lower-dimensional manifold Ω0 Thenfor forall119911 = (1199111 119911119899) isin Ω Ω0 we have120597120588 (119911)120597119911 = (120597120588 (119911)1205971199111 120597120588 (119911)1205971199112 120597120588 (119911)120597119911119899 )

120597120588 (119911)120597119911119895 119911119895 ge 0120588 (119911) = 2 119899sum

119895=1

120597120588 (119911)120597119911119895 119911119895(65)

Theorem 19 Let 119891(119911) be an almost starlike function of com-plex order 120582 on 119863 with 120582 isin C and Re 120582 le 0 Let Ω sub C119899 be abounded and completely Reinhardt domain whose Minkowskifunctional 120588(119911) is 1198621 on Ω 0 Let

119865 (119911) = (119903119891(1199111119903 ) (119903119891 (1199111119903)1199111 )1205732sdot 1199112 (119903119891 (1199111119903)1199111 )120573119899 119911119899)1015840

(66)

where 120573119895 isin [0 1] 119903 = sup|1199111| 119911 = (1199111 119911119899)1015840 isin Ω and(119903119891(1199111119903)1199111)120573119895 |1199111=0 = 1 (119895 = 2 119899) Then 119865(119911) is an almoststarlike mapping of complex order 120582 on ΩProof For

119865 (119911) = (119903119891(1199111119903 ) (119903119891 (1199111119903)1199111 )1205732sdot 1199112 (119903119891 (1199111119903)1199111 )120573119899 119911119899)1015840

(67)

immediately we have

119869119865 (119911) =(((((((((((

1198911015840 (1199111119903 ) 0 sdot sdot sdot 01205732119903 [1198911015840 (1199111119903)119891 (1199111119903) minus 1199031199111 ](119903119891 (1199111119903)1199111 )1205732 1199112 (119903119891 (1199111119903)1199111 )1205732 sdot sdot sdot 0

sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot120573119899119903 [1198911015840 (1199111119903)119891 (1199111119903) minus 1199031199111](119903119891 (1199111119903)1199111 )120573119899 119911119899 0 sdot sdot sdot (119903119891 (1199111119903)1199111 )120573119899

)))))))))))

(68)

Therefore

119869minus1119865 (119911) =(((((((((((

11198911015840 (1199111119903) 0 sdot sdot sdot 01205732119903 [ 11990311991111198911015840 (1199111119903) minus 1119891 (1199111119903)] 1199112 (119903119891 (1199111119903)1199111 )minus1205732 sdot sdot sdot 0

sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot120573119899119903 [ 11990311991111198911015840 (1199111119903) minus 1119891 (1199111119903)] 119911119899 0 sdot sdot sdot (119903119891 (1199111119903)1199111 )minus120573119899

)))))))))))

(69)

Discrete Dynamics in Nature and Society 9

Then 119869minus1119865 (119911) 119865 (119911)= (119903119891 (1199111119903)1198911015840 (1199111119903) (1 minus 1205732 + 1205732 119903119891 (1199111119903)11991111198911015840 (1199111119903))sdot 1199112 (1 minus 120573119899 + 120573119899 119903119891 (1199111119903)11991111198911015840 (1199111119903)) 119911119899)1015840

(70)

By Lemma 18 we get(1 minus 120582) 2120597120588120597119911 (119911) 119869minus1119865 (119911) 119865 (119911) = 2 (1 minus 120582)sdot [[ 119903119891 (1199111119903)11991111198911015840 (1199111119903) 120597120588 (119911)1205971199111 1199111+ 119899sum119895=2

(1 minus 120573119895 + 120573119895 119903119891 (1199111119903)11991111198911015840 (1199111119903)) 120597120588 (119911)120597119911119895 119911119895]] (71)

Since119891(119911) is an almost starlike function of complex order120582 thenR[(1 minus 120582) 119903119891 (1199111119903)11991111198911015840 (1199111119903)] ge minusR120582 (72)

Therefore by Lemma 18 we obtain

R [(1 minus 120582) 2120597120588120597119911 (119911) 119869minus1119865 (119911) 119865 (119911)] = R[(1 minus 120582)sdot 119903119891 (1199111119903)11991111198911015840 (1199111119903)] 2120597120588 (119911)1205971199111 1199111 + 119899sum

119895=2

(1 minus 120573119895)R (1 minus 120582)+ 120573119895R[(1 minus 120582) 119903119891 (1199111119903)11991111198911015840 (1199111119903)] 2120597120588 (119911)120597119911119895 119911119895 ge minusR120582sdot 2120597120588 (119911)1205971199111 1199111 + 119899sum

119895=2

(1 minus 120573119895) (1 minusR120582) + 120573119895 (minusR120582)sdot 2120597120588 (119911)120597119911119895 119911119895 = (minusR120582) [2120597120588 (119911)1205971199111 1199111 + 2120597120588 (119911)120597119911119895 119911119895]+ 119899sum119895=2

(1 minus 120573119895) 2120597120588 (119911)120597119911119895 119911119895 ge (minusR120582) 120588 (119911)

(73)

Hence 119865(119911) is an almost starlike mapping of complex order 120582onΩ by Definition 4

Theorem 20 Let 119891119895 (119895 = 1 2 119899) be almost starlikefunctions of complex order 120582 on 119863 with 120582 isin C and Re 120582 lt 0Let Ω sub C119899 be a bounded starlike and circular domain whoseMinkowski functional 120588(119911) is1198621 onΩ0 Let 119903119895 be the radiusof the disk 119880119895 = 119911119895 isin C 119911 = (1199111 119911119895 119911119899)1015840 isin Ω whosecenter is zero Let

119865 (119911) = 119911 119899prod119895=1

(119903119895119891119895 (119911119895119903119895)119911119895 )120582119894 (74)

where 120582119894119895 ge 0 sum119899119895=1 120582119894119895 = 1 and (119891119895(119911119895)119911119895)120582119894119895 |119911119895=0 = 1 (119894 119895 =1 2 119899)Then119865(119911) is an almost starlikemapping of complexorder 120582 on ΩProof Let

120577119895 = 119911119895119903119895 119892 (119911) = 119899prod

119895=1

(119891119895 (120577119895)120577119895 )120582119894 (75)

which follows 119865(119911) = 119911119892(119911) Then 119869119865(119911) = 119869119892(119911)119911 + 119892(119911) Bydirect computation we get

119869119892 (119911) 119911 = 119892 (119911) [[119899sum119895=1

1205821198951198911015840119895 (119911119895119903119895) 119911119895119903119895119891119895 (119911119895119903119895) minus 1]]= 119892 (119911) [[

119899sum119895=1

1205821198951198911015840119895 (120577119895) 120577119895119891119895 (120577119895) minus 1]] (76)

Since 119891119895 are almost starlike functions of complex order 120582 on119863 then

R[[(1 minus 120582) 119891119895 (120577119895)1205771198951198911015840119895 (120577119895)]] ge minusR120582 gt 0 (77)

Therefore

R[ 11 minus 120582 (1 + 119869119892 (119911) 119911119892 (119911) )]= R[[ 11 minus 120582 119899sum

119895=1

1205821198951198911015840119895 (120577119895) 120577119895119891119895 (120577119895) ]]= 119899sum119895=1

120582119895R[[ 11 minus 120582 1198911015840119895 (120577119895) 120577119895119891119895 (120577119895) ]] gt 0(78)

Then 119869119892(119911)119911+119892(119911) = 0 It is obvious that119865(0) = 0 and 119869119865(0) =119868 so 119865(119911) is a normalized locally biholomorphic mapping onΩ and

119869minus1119865 (119911) 119865 (119911) = 119892 (119911) 119911119869119892 (119911) 119911 + 119892 (119911) (79)

Let

119902119895 (120577119895) = 2Re (minus120582) 1198911015840119895 (120577119895) 120577119895(1 minus 120582) 119891119895 (120577119895) minus 1 (80)

(77) follows 100381610038161003816100381610038161003816100381610038161003816100381610038162Re (minus120582) 1198911015840119895 (120577119895) 120577119895(1 minus 120582) 119891119895 (120577119895) minus 110038161003816100381610038161003816100381610038161003816100381610038161003816 le 1 (81)

10 Discrete Dynamics in Nature and Society

that is |119902119895(120577119895)| le 1 For Re 120582 lt 0 by (76) (79) and Lemma 18we obtain1003816100381610038161003816100381610038161003816100381610038162R (minus120582) 1(1 minus 120582) (2120588 (119911)) (120597120588120597119911) (119911) 119869minus1119865 (119911) 119865 (119911)

minus 1100381610038161003816100381610038161003816100381610038161003816 =100381610038161003816100381610038161003816100381610038161003816100381610038162R (minus120582) 11 minus 120582 119899sum

119895=1

1205821198951198911015840119895 (120577119895) 120577119895119891119895 (120577119895) minus 110038161003816100381610038161003816100381610038161003816100381610038161003816= 10038161003816100381610038161003816100381610038161003816100381610038161003816 119899sum119895=1120582119895119902119895 (120577119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816 le 119899sum119895=1120582119895 = 1(82)

which leads to

R [(1 minus 120582) 2120597120588120597119911 (119911) 119869minus1119865 (119911) 119865 (119911)] ge minusR120582120588 (119911) (83)

Hence 119865(119911) is an almost starlike mapping of complex order 120582onΩ by Definition 4

Similar to Theorem 20 we can get the following conclu-sion

Theorem 21 Let 119891119895 (119895 = 1 2 119899) be almost starlikefunctions of complex order 120582 on 119863 with 120582 isin C and Re 120582 lt 0Let

119865 (119909) = 119909 119899prod119895=1

(119891119895 (119879119906119895 (119909))119879119906119895 (119909) )120582119895 119909 isin 119861 (84)

where 120582119894119895 ge 0 sum119899119895=1 120582119894119895 = 1 (119891119895(119909119895)119909119895)120582119894119895 |119909119895=0 = 1 (119909119895 =119879119906119895(119909) 119894 119895 = 1 2 119899) and 119906119895 is the unit vector of thecomplex Banach space 119883 Then 119865(119909) is an almost starlikemapping of complex order 120582 on the unit ball 119861 in complexBanach space119883

Theorem 22 Let 119891 be an almost starlike function of complexorder 120582 on119863 with 120582 isin C and Re 120582 le 0 Let

119865 (119909) = 119899minus1sum119895=1

(119891(1198791199091 (119909))1198791199091 (119909) )120573119895 119879119909119895 (119909) 119909119895 + 119909minus 119899minus1sum119895=1

119879119909119895 (119909) 119909119895 119909 isin 119861 (85)

where 1199091 isin 119861 1199091 = 1 120573119895 isin [0 1] (119891(119911)119911)120573119895 |119911=0 = 1 (119895 =1 2 119899 minus 1) and 1199091 119909119899 are linearly independent and119879119909119894 isin 119883lowast such that 119879119909119894 = 1 119879119909119894(119909119894) = 1 and 119879119909119894(119909119895) =0 (119894 = 119895) for forall119909119894 Then 119865(119909) is an almost starlike mapping ofcomplex order 120582 on the unit ball 119861 in complex Banach space119883

Proof From [24] we get 119865(119909) is a normalized biholomorphicmapping on 119861 and119909 119879119909 [119863119865 (119909)minus1 119865 (119909)]= 1199092

+ 119899minus1sum119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162 [ 119891 (1198791199091 (119909))1198791199091 (119909) 1198911015840 (1198791199091 (119909)) minus 1] (86)

Since119891 is an almost starlike function of complex order120582 thenR[[(1 minus 120582) 119891 (119879119909119895 (119909))119879119909119895 (119909) 1198911015840 (119879119909119895 (119909))]] ge minusR120582 (87)

Therefore

R (1 minus 120582) 119879119909 [119863119865 (119909)minus1 119865 (119909)] = (1 minusR120582) 119909+ 119899minus1sum119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162119909sdotR(1 minus 120582) [ 119891 (1198791199091 (119909))1198791199091 (119909) 1198911015840 (1198791199091 (119909)) minus 1]ge (1 minusR120582) 119909 + 119899minus1sum

119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162119909sdot [R (minus120582) minus (1 minusR120582)] = minusR120582 119909+ 1119909 [[1199092 minus

119899minus1sum119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162]] ge minusR120582 119909

(88)

which lead to the desired conclusion by Definition 5

Similar to Theorem 22 we can get the following conclu-sion

Theorem 23 Let 119891 be an almost starlike function of complexorder 120582 on119863 with 120582 isin C Re 120582 le 0 Let119865 (119909) = Φ120573 (119891) (119909)= 119891 (1198791199091 (119909)) 1199091

+ (119891 (1198791199091 (119909))1198791199091 (119909) )120573 (119909 minus 1198791199091 (119909) 1199091) (89)

where 0 le 120573 le 1 1199091 isin 119861 1199091 = 1 (119891(1199111)1199111)120573|1199111=0 = 1 and119891(1199111) = 0 for 1199111 = 0 Then 119865(119909) is an almost starlike mappingof complex order 120582 on 119861Remark 24 Setting 120582 = 120572(120572 minus 1) in Theorems 19ndash23 weget the corresponding results with respect to almost starlikemappings of order 120572

Discrete Dynamics in Nature and Society 11

Competing Interests

The authors declare that they have no conflict of interests

Acknowledgments

This work is supported by NSF of China (no U1204618) andScience and Technology Research Projects of Henan Provin-cial Education Department (nos 14B110015 and 14B110016)

References

[1] C Pommerenke Univalent Functions Vandenhoeck amp Ru-precht Gottingen Germany 1975

[2] H Cartan ldquoSur la possibilite drsquoentendre aux functions de plus-ieurs variables complexes la theorie des functions univalentsrdquoin Lecons sur les Functions Univalents onMutivalents P MontelEd Gauthier-Villar Paris France 1933

[3] R W Barnard C H FitzGerald and S Gong ldquoA distortionthoerem for biholomorphicmappings inC2rdquoTransactions of theAmerican Mathematical Society vol 344 pp 902ndash924 1994

[4] T S Liu andW J Zhang ldquoA distortion thoerem for biholomor-phic convex mappings in C119899rdquo Chinese Journal of ContemporaryMathematics vol 20 no 3 pp 421ndash430 1999

[5] S Gong S K Wang and Q H Yu ldquoBiholomorphic convexmappings of ball in C119899rdquo Pacific Journal of Mathematics vol 161no 2 pp 287ndash306 1993

[6] S Gong and T S Liu ldquoDistortion theorems for biholomorphicconvex mappings on bounded convex circular domainsrdquo Chi-nese Annals of Mathematics Series B vol 20 no 3 pp 297ndash3041999

[7] I Graham and D Varolin ldquoBloch constants in one and severalvariablesrdquoPacific Journal ofMathematics vol 174 no 2 pp 347ndash357 1996

[8] P L DurenUnivalent Functions Springer New York NY USA1983

[9] T H Gronwall ldquoSome remarks on conformal representationrdquoAnnals of Mathematics vol 16 no 1ndash4 pp 72ndash76 191415

[10] M I Robertson ldquoOn the theory of univalent functionsrdquo Annalsof Mathematics vol 37 no 2 pp 374ndash408 1936

[11] A V Boyd ldquoCoefficient estimates for starlike functions of order120572rdquo Proceedings of the AmericanMathematical Society vol 17 pp1016ndash1018 1966

[12] H Silverman ldquoUnivalent functions with negative coefficientsrdquoProceedings of the American Mathematical Society vol 51 pp109ndash116 1975

[13] H M Srivastava and S Owa ldquoCertain subclasses of starlikefunctions Irdquo Journal ofMathematical Analysis and Applicationsvol 161 no 2 pp 405ndash415 1991

[14] K A Roper and T J Suffridge ldquoConvex mappings on the unitball ofC119899rdquo Journal drsquoAnalyseMathematique vol 65 pp 333ndash3471995

[15] I Graham and G Kohr ldquoUnivalent mappings associated withthe Roper-Suffridge extension operatorrdquo Journal drsquoAnalyseMathematique vol 81 pp 331ndash342 2000

[16] S Gong and T S Liu ldquoThe generalized Roper-Suffridge exten-sion operatorrdquo Journal of Mathematical Analysis and Applica-tions vol 284 no 2 pp 425ndash434 2003

[17] G Kohr ldquoLoewner chains and a modification of the Roper-Suffridge extension operatorrdquo Mathematica vol 71 no 1 pp41ndash48 2006

[18] J R Muir ldquoA class of Loewner chain preserving extensionoperatorsrdquo Journal of Mathematical Analysis and Applicationsvol 337 no 2 pp 862ndash879 2008

[19] Y H Zhao Almost Starlike Mappings of Complex Order 120582 onthe Unit Ball of a Complex Banach Space Zhejiang NormalUniversity Jinhua China 2013

[20] L V Ahlfors Complex Analysis McGraw-Hill New York NYUSA 3rd edition 1978

[21] L Bieberbach ldquoUber einige extremal problem in gebiete derkonformen abbildungrdquo Mathematische Annalen vol 77 no 2pp 153ndash172 1916

[22] T Hayami S Owa and H M Srivastava ldquoCoefficient inequal-ities for certain classes of analytic and univalent functionsrdquoJournal of Inequalities in Pure and Applied Mathematics vol 8no 4 pp 1ndash10 2007

[23] M S Liu and Y C Zhu ldquoGeneralized Roper-Suffridge oper-ators on bounded and complete Reinhardt domainsrdquo Sciencein China vol 37 no 10 pp 1193ndash1206 2007

[24] Y Y Cui and C J Wang ldquoThe generalized Roper-Suffridgeextension operators in complex Banach spacerdquo Journal ofMathematics vol 34 no 3 pp 515ndash520 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 7

Lemma 13 (see [22]) Let119901(119911) = 1+suminfin119899=1 119888119899119911119899 be holomorphicon119863 ThenR119901(119911) gt 0 if and only if

119901 (119911) = 120585 minus 1120585 + 1 120585 isin 120597119863 (48)

Theorem 14 Let 119891(119911) = 119911 + suminfin119899=2 119886119899119911119899 be an almost starlikefunction of complex order 120582 with 120582 isin C Re 120582 le 0 on119863 Then10038161003816100381610038161198861198991003816100381610038161003816 le 119899minus1prod

119895=2

( 2|1 minus 120582| + 119895 minus 2119895 minus 1) 119899 = 2 3 (49)

Proof Since 119891(119911) is an almost starlike function of complexorder 120582 then

R [(1 minus 120582) 119891 (119911)1199111198911015840 (119911) + 120582] gt 0 119911 isin 119863 (50)

Let 119901(119911) = (1 minus 120582)(119891(119911)1199111198911015840(119911)) + 120582 Then we have1199111198911015840 (119911) (119901 (119911) minus 120582) = (1 minus 120582) 119891 (119911) (51)

and 119901(0) = 1 andR119901(119911) gt 0 Let 119901(119911) = 1 + suminfin119899=1 119888119899119911119899 Then10038161003816100381610038161198881198991003816100381610038161003816 le 2 (119899 = 1 2 ) (52)

by Lemma 12 For 119891(119911) = 119911 + suminfin119899=2 119886119899119911119899 (51) follows119911(1 + infinsum

119899=2

119899119886119899119911119899minus1)(1 minus 120582 + infinsum119899=1

119888119899119911119899)= (1 minus 120582)(119911 + infinsum

119899=2

119886119899119911119899) (53)

Therefore(1 minus 120582) 119911 + [1198881 + 21198862 (1 minus 120582)] 1199112 + sdot sdot sdot + [119888119899minus1+ 21198862119888119899minus2 + sdot sdot sdot + (119899 minus 1) 119886119899minus11198881 + 119899119886119899 (1 minus 120582)] 119911119899+ sdot sdot sdot = (1 minus 120582) 119911 + 1198862 (1 minus 120582) 1199112 + sdot sdot sdot + 119886119899 (1 minus 120582)sdot 119911119899 + sdot sdot sdot (54)

So 119886119899 = 119888119899minus1 + 21198862119888119899minus2 + sdot sdot sdot + (119899 minus 1) 119886119899minus11198881(119899 minus 1) (120582 minus 1) (55)

By (52) we get10038161003816100381610038161198861198991003816100381610038161003816 le 2 [1 + 2 100381610038161003816100381611988621003816100381610038161003816 + sdot sdot sdot + (119899 minus 1) 1003816100381610038161003816119886119899minus11003816100381610038161003816](119899 minus 1) |1 minus 120582| (56)

Bymathematical induction we obtain the desired conclusion

Theorem 15 Let 119891(119911) be an almost starlike function ofcomplex order 120582 with 120582 isin C Re 120582 le 0 on 119863 and let 119911 = 0be the zero of order 119896 + 1 (119896 isin N) of 119891(119911) minus 119911 Let 119891(119911) =119911 + suminfin119899=119896+1 119886119899119911119899 Then1003816100381610038161003816119886119899+11003816100381610038161003816 le 2|120582 + 119899| (119899 = 119896 119896 + 1 ) (57)

Proof Since 119891(119911) = 119911 + suminfin119899=119896+1 119886119899119911119899 similar to Theorem 14by (51) we obtain

119911(1 + infinsum119899=119896+1

119899119886119899119911119899minus1)(1 minus 120582 + infinsum119899=1

119888119899119911119899)= (1 minus 120582)(119911 + infinsum

119899=119896+1

119886119899119911119899) (58)

which follows(1 minus 120582) 119911 + 11988811199112 + sdot sdot sdot + 119888119896minus1119911119896+ [119888119896 + (119896 + 1) 119886119896+1] 119911119896+1 + sdot sdot sdot+ [119888119899 + (119899 + 1) 119886119899+1] 119911119899+1 + sdot sdot sdot= (1 minus 120582) 119911 + (1 minus 120582) 119886119896+1119911119896+1 + sdot sdot sdot+ (1 minus 120582) 119886119899+1119911119899+1 + sdot sdot sdot (59)

Therefore 1198881 = sdot sdot sdot = 119888119896minus1 = 0119888119899 = (minus120582 minus 119899) 119886119899+1 (119899 = 119896 119896 + 1 ) (60)

By (52) we get the desired conclusion

Theorem 16 119891(119911) = 119911 + suminfin119899=2 119886119899119911119899 is an almost starlikefunction of complex order 120582 with 120582 isin C and Re 120582 le 0 on 119863if and only ifinfinsum119899=2

[(1 + (119899 minus 1) 120582) (120585 + 1) + 119899 (1 minus 120585)] 119886119899119911119899 = minus2(120585 isin 120597119863 119899 = 2 3 ) (61)

Proof Let 119901(119911) = (1 minus 120582)(119891(119911)1199111198911015840(119911)) + 120582 Since 119891(119911) isan almost starlike function of complex order 120582 then 119901(119911)is holomorphic on 119863 and 119901(0) = 0 and R119901(119911) gt 0 ByLemma 13 we get(1 minus 120582) 119891 (119911)1199111198911015840 (119911) + 120582 = 120585 minus 1120585 + 1 120585 isin 120597119863 (62)

which follows(1 minus 120582) (120585 + 1) 119891 (119911) + 120582 (120585 + 1) 1199111198911015840 (119911)= (120585 minus 1) 1199111198911015840 (119911) (63)

For 119891(119911) = 119911 + suminfin119899=2 119886119899119911119899 we have(1 minus 120582) (120585 + 1) (119911 + infinsum119899=2

119886119899119911119899)+ 120582 (120585 + 1) (119911 + infinsum

119899=2

119899119886119899119911119899)= (120585 minus 1) (119911 + infinsum

119899=2

119899119886119899119911119899) (64)

which leads to the desired conclusion

8 Discrete Dynamics in Nature and Society

Remark 17 Setting 120582 = 120572(120572 minus 1) in Theorems 14ndash16 weget the corresponding results for almost starlike functions oforder 1205724 The Invariance of Almost Starlike Mappings

of Complex Order 120582In this section we mainly discuss the invariance of almoststarlike mappings of complex order 120582 on Reinhardt domainsand on the unit ball in complex Banach spaces under somegeneralized Roper-Suffridge operators

Lemma 18 (see [23]) Let Ω sub C119899 be a bounded andcompletely Reinhardt domain whose Minkowski functional120588(119911) is 1198621 except for a lower-dimensional manifold Ω0 Thenfor forall119911 = (1199111 119911119899) isin Ω Ω0 we have120597120588 (119911)120597119911 = (120597120588 (119911)1205971199111 120597120588 (119911)1205971199112 120597120588 (119911)120597119911119899 )

120597120588 (119911)120597119911119895 119911119895 ge 0120588 (119911) = 2 119899sum

119895=1

120597120588 (119911)120597119911119895 119911119895(65)

Theorem 19 Let 119891(119911) be an almost starlike function of com-plex order 120582 on 119863 with 120582 isin C and Re 120582 le 0 Let Ω sub C119899 be abounded and completely Reinhardt domain whose Minkowskifunctional 120588(119911) is 1198621 on Ω 0 Let

119865 (119911) = (119903119891(1199111119903 ) (119903119891 (1199111119903)1199111 )1205732sdot 1199112 (119903119891 (1199111119903)1199111 )120573119899 119911119899)1015840

(66)

where 120573119895 isin [0 1] 119903 = sup|1199111| 119911 = (1199111 119911119899)1015840 isin Ω and(119903119891(1199111119903)1199111)120573119895 |1199111=0 = 1 (119895 = 2 119899) Then 119865(119911) is an almoststarlike mapping of complex order 120582 on ΩProof For

119865 (119911) = (119903119891(1199111119903 ) (119903119891 (1199111119903)1199111 )1205732sdot 1199112 (119903119891 (1199111119903)1199111 )120573119899 119911119899)1015840

(67)

immediately we have

119869119865 (119911) =(((((((((((

1198911015840 (1199111119903 ) 0 sdot sdot sdot 01205732119903 [1198911015840 (1199111119903)119891 (1199111119903) minus 1199031199111 ](119903119891 (1199111119903)1199111 )1205732 1199112 (119903119891 (1199111119903)1199111 )1205732 sdot sdot sdot 0

sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot120573119899119903 [1198911015840 (1199111119903)119891 (1199111119903) minus 1199031199111](119903119891 (1199111119903)1199111 )120573119899 119911119899 0 sdot sdot sdot (119903119891 (1199111119903)1199111 )120573119899

)))))))))))

(68)

Therefore

119869minus1119865 (119911) =(((((((((((

11198911015840 (1199111119903) 0 sdot sdot sdot 01205732119903 [ 11990311991111198911015840 (1199111119903) minus 1119891 (1199111119903)] 1199112 (119903119891 (1199111119903)1199111 )minus1205732 sdot sdot sdot 0

sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot120573119899119903 [ 11990311991111198911015840 (1199111119903) minus 1119891 (1199111119903)] 119911119899 0 sdot sdot sdot (119903119891 (1199111119903)1199111 )minus120573119899

)))))))))))

(69)

Discrete Dynamics in Nature and Society 9

Then 119869minus1119865 (119911) 119865 (119911)= (119903119891 (1199111119903)1198911015840 (1199111119903) (1 minus 1205732 + 1205732 119903119891 (1199111119903)11991111198911015840 (1199111119903))sdot 1199112 (1 minus 120573119899 + 120573119899 119903119891 (1199111119903)11991111198911015840 (1199111119903)) 119911119899)1015840

(70)

By Lemma 18 we get(1 minus 120582) 2120597120588120597119911 (119911) 119869minus1119865 (119911) 119865 (119911) = 2 (1 minus 120582)sdot [[ 119903119891 (1199111119903)11991111198911015840 (1199111119903) 120597120588 (119911)1205971199111 1199111+ 119899sum119895=2

(1 minus 120573119895 + 120573119895 119903119891 (1199111119903)11991111198911015840 (1199111119903)) 120597120588 (119911)120597119911119895 119911119895]] (71)

Since119891(119911) is an almost starlike function of complex order120582 thenR[(1 minus 120582) 119903119891 (1199111119903)11991111198911015840 (1199111119903)] ge minusR120582 (72)

Therefore by Lemma 18 we obtain

R [(1 minus 120582) 2120597120588120597119911 (119911) 119869minus1119865 (119911) 119865 (119911)] = R[(1 minus 120582)sdot 119903119891 (1199111119903)11991111198911015840 (1199111119903)] 2120597120588 (119911)1205971199111 1199111 + 119899sum

119895=2

(1 minus 120573119895)R (1 minus 120582)+ 120573119895R[(1 minus 120582) 119903119891 (1199111119903)11991111198911015840 (1199111119903)] 2120597120588 (119911)120597119911119895 119911119895 ge minusR120582sdot 2120597120588 (119911)1205971199111 1199111 + 119899sum

119895=2

(1 minus 120573119895) (1 minusR120582) + 120573119895 (minusR120582)sdot 2120597120588 (119911)120597119911119895 119911119895 = (minusR120582) [2120597120588 (119911)1205971199111 1199111 + 2120597120588 (119911)120597119911119895 119911119895]+ 119899sum119895=2

(1 minus 120573119895) 2120597120588 (119911)120597119911119895 119911119895 ge (minusR120582) 120588 (119911)

(73)

Hence 119865(119911) is an almost starlike mapping of complex order 120582onΩ by Definition 4

Theorem 20 Let 119891119895 (119895 = 1 2 119899) be almost starlikefunctions of complex order 120582 on 119863 with 120582 isin C and Re 120582 lt 0Let Ω sub C119899 be a bounded starlike and circular domain whoseMinkowski functional 120588(119911) is1198621 onΩ0 Let 119903119895 be the radiusof the disk 119880119895 = 119911119895 isin C 119911 = (1199111 119911119895 119911119899)1015840 isin Ω whosecenter is zero Let

119865 (119911) = 119911 119899prod119895=1

(119903119895119891119895 (119911119895119903119895)119911119895 )120582119894 (74)

where 120582119894119895 ge 0 sum119899119895=1 120582119894119895 = 1 and (119891119895(119911119895)119911119895)120582119894119895 |119911119895=0 = 1 (119894 119895 =1 2 119899)Then119865(119911) is an almost starlikemapping of complexorder 120582 on ΩProof Let

120577119895 = 119911119895119903119895 119892 (119911) = 119899prod

119895=1

(119891119895 (120577119895)120577119895 )120582119894 (75)

which follows 119865(119911) = 119911119892(119911) Then 119869119865(119911) = 119869119892(119911)119911 + 119892(119911) Bydirect computation we get

119869119892 (119911) 119911 = 119892 (119911) [[119899sum119895=1

1205821198951198911015840119895 (119911119895119903119895) 119911119895119903119895119891119895 (119911119895119903119895) minus 1]]= 119892 (119911) [[

119899sum119895=1

1205821198951198911015840119895 (120577119895) 120577119895119891119895 (120577119895) minus 1]] (76)

Since 119891119895 are almost starlike functions of complex order 120582 on119863 then

R[[(1 minus 120582) 119891119895 (120577119895)1205771198951198911015840119895 (120577119895)]] ge minusR120582 gt 0 (77)

Therefore

R[ 11 minus 120582 (1 + 119869119892 (119911) 119911119892 (119911) )]= R[[ 11 minus 120582 119899sum

119895=1

1205821198951198911015840119895 (120577119895) 120577119895119891119895 (120577119895) ]]= 119899sum119895=1

120582119895R[[ 11 minus 120582 1198911015840119895 (120577119895) 120577119895119891119895 (120577119895) ]] gt 0(78)

Then 119869119892(119911)119911+119892(119911) = 0 It is obvious that119865(0) = 0 and 119869119865(0) =119868 so 119865(119911) is a normalized locally biholomorphic mapping onΩ and

119869minus1119865 (119911) 119865 (119911) = 119892 (119911) 119911119869119892 (119911) 119911 + 119892 (119911) (79)

Let

119902119895 (120577119895) = 2Re (minus120582) 1198911015840119895 (120577119895) 120577119895(1 minus 120582) 119891119895 (120577119895) minus 1 (80)

(77) follows 100381610038161003816100381610038161003816100381610038161003816100381610038162Re (minus120582) 1198911015840119895 (120577119895) 120577119895(1 minus 120582) 119891119895 (120577119895) minus 110038161003816100381610038161003816100381610038161003816100381610038161003816 le 1 (81)

10 Discrete Dynamics in Nature and Society

that is |119902119895(120577119895)| le 1 For Re 120582 lt 0 by (76) (79) and Lemma 18we obtain1003816100381610038161003816100381610038161003816100381610038162R (minus120582) 1(1 minus 120582) (2120588 (119911)) (120597120588120597119911) (119911) 119869minus1119865 (119911) 119865 (119911)

minus 1100381610038161003816100381610038161003816100381610038161003816 =100381610038161003816100381610038161003816100381610038161003816100381610038162R (minus120582) 11 minus 120582 119899sum

119895=1

1205821198951198911015840119895 (120577119895) 120577119895119891119895 (120577119895) minus 110038161003816100381610038161003816100381610038161003816100381610038161003816= 10038161003816100381610038161003816100381610038161003816100381610038161003816 119899sum119895=1120582119895119902119895 (120577119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816 le 119899sum119895=1120582119895 = 1(82)

which leads to

R [(1 minus 120582) 2120597120588120597119911 (119911) 119869minus1119865 (119911) 119865 (119911)] ge minusR120582120588 (119911) (83)

Hence 119865(119911) is an almost starlike mapping of complex order 120582onΩ by Definition 4

Similar to Theorem 20 we can get the following conclu-sion

Theorem 21 Let 119891119895 (119895 = 1 2 119899) be almost starlikefunctions of complex order 120582 on 119863 with 120582 isin C and Re 120582 lt 0Let

119865 (119909) = 119909 119899prod119895=1

(119891119895 (119879119906119895 (119909))119879119906119895 (119909) )120582119895 119909 isin 119861 (84)

where 120582119894119895 ge 0 sum119899119895=1 120582119894119895 = 1 (119891119895(119909119895)119909119895)120582119894119895 |119909119895=0 = 1 (119909119895 =119879119906119895(119909) 119894 119895 = 1 2 119899) and 119906119895 is the unit vector of thecomplex Banach space 119883 Then 119865(119909) is an almost starlikemapping of complex order 120582 on the unit ball 119861 in complexBanach space119883

Theorem 22 Let 119891 be an almost starlike function of complexorder 120582 on119863 with 120582 isin C and Re 120582 le 0 Let

119865 (119909) = 119899minus1sum119895=1

(119891(1198791199091 (119909))1198791199091 (119909) )120573119895 119879119909119895 (119909) 119909119895 + 119909minus 119899minus1sum119895=1

119879119909119895 (119909) 119909119895 119909 isin 119861 (85)

where 1199091 isin 119861 1199091 = 1 120573119895 isin [0 1] (119891(119911)119911)120573119895 |119911=0 = 1 (119895 =1 2 119899 minus 1) and 1199091 119909119899 are linearly independent and119879119909119894 isin 119883lowast such that 119879119909119894 = 1 119879119909119894(119909119894) = 1 and 119879119909119894(119909119895) =0 (119894 = 119895) for forall119909119894 Then 119865(119909) is an almost starlike mapping ofcomplex order 120582 on the unit ball 119861 in complex Banach space119883

Proof From [24] we get 119865(119909) is a normalized biholomorphicmapping on 119861 and119909 119879119909 [119863119865 (119909)minus1 119865 (119909)]= 1199092

+ 119899minus1sum119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162 [ 119891 (1198791199091 (119909))1198791199091 (119909) 1198911015840 (1198791199091 (119909)) minus 1] (86)

Since119891 is an almost starlike function of complex order120582 thenR[[(1 minus 120582) 119891 (119879119909119895 (119909))119879119909119895 (119909) 1198911015840 (119879119909119895 (119909))]] ge minusR120582 (87)

Therefore

R (1 minus 120582) 119879119909 [119863119865 (119909)minus1 119865 (119909)] = (1 minusR120582) 119909+ 119899minus1sum119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162119909sdotR(1 minus 120582) [ 119891 (1198791199091 (119909))1198791199091 (119909) 1198911015840 (1198791199091 (119909)) minus 1]ge (1 minusR120582) 119909 + 119899minus1sum

119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162119909sdot [R (minus120582) minus (1 minusR120582)] = minusR120582 119909+ 1119909 [[1199092 minus

119899minus1sum119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162]] ge minusR120582 119909

(88)

which lead to the desired conclusion by Definition 5

Similar to Theorem 22 we can get the following conclu-sion

Theorem 23 Let 119891 be an almost starlike function of complexorder 120582 on119863 with 120582 isin C Re 120582 le 0 Let119865 (119909) = Φ120573 (119891) (119909)= 119891 (1198791199091 (119909)) 1199091

+ (119891 (1198791199091 (119909))1198791199091 (119909) )120573 (119909 minus 1198791199091 (119909) 1199091) (89)

where 0 le 120573 le 1 1199091 isin 119861 1199091 = 1 (119891(1199111)1199111)120573|1199111=0 = 1 and119891(1199111) = 0 for 1199111 = 0 Then 119865(119909) is an almost starlike mappingof complex order 120582 on 119861Remark 24 Setting 120582 = 120572(120572 minus 1) in Theorems 19ndash23 weget the corresponding results with respect to almost starlikemappings of order 120572

Discrete Dynamics in Nature and Society 11

Competing Interests

The authors declare that they have no conflict of interests

Acknowledgments

This work is supported by NSF of China (no U1204618) andScience and Technology Research Projects of Henan Provin-cial Education Department (nos 14B110015 and 14B110016)

References

[1] C Pommerenke Univalent Functions Vandenhoeck amp Ru-precht Gottingen Germany 1975

[2] H Cartan ldquoSur la possibilite drsquoentendre aux functions de plus-ieurs variables complexes la theorie des functions univalentsrdquoin Lecons sur les Functions Univalents onMutivalents P MontelEd Gauthier-Villar Paris France 1933

[3] R W Barnard C H FitzGerald and S Gong ldquoA distortionthoerem for biholomorphicmappings inC2rdquoTransactions of theAmerican Mathematical Society vol 344 pp 902ndash924 1994

[4] T S Liu andW J Zhang ldquoA distortion thoerem for biholomor-phic convex mappings in C119899rdquo Chinese Journal of ContemporaryMathematics vol 20 no 3 pp 421ndash430 1999

[5] S Gong S K Wang and Q H Yu ldquoBiholomorphic convexmappings of ball in C119899rdquo Pacific Journal of Mathematics vol 161no 2 pp 287ndash306 1993

[6] S Gong and T S Liu ldquoDistortion theorems for biholomorphicconvex mappings on bounded convex circular domainsrdquo Chi-nese Annals of Mathematics Series B vol 20 no 3 pp 297ndash3041999

[7] I Graham and D Varolin ldquoBloch constants in one and severalvariablesrdquoPacific Journal ofMathematics vol 174 no 2 pp 347ndash357 1996

[8] P L DurenUnivalent Functions Springer New York NY USA1983

[9] T H Gronwall ldquoSome remarks on conformal representationrdquoAnnals of Mathematics vol 16 no 1ndash4 pp 72ndash76 191415

[10] M I Robertson ldquoOn the theory of univalent functionsrdquo Annalsof Mathematics vol 37 no 2 pp 374ndash408 1936

[11] A V Boyd ldquoCoefficient estimates for starlike functions of order120572rdquo Proceedings of the AmericanMathematical Society vol 17 pp1016ndash1018 1966

[12] H Silverman ldquoUnivalent functions with negative coefficientsrdquoProceedings of the American Mathematical Society vol 51 pp109ndash116 1975

[13] H M Srivastava and S Owa ldquoCertain subclasses of starlikefunctions Irdquo Journal ofMathematical Analysis and Applicationsvol 161 no 2 pp 405ndash415 1991

[14] K A Roper and T J Suffridge ldquoConvex mappings on the unitball ofC119899rdquo Journal drsquoAnalyseMathematique vol 65 pp 333ndash3471995

[15] I Graham and G Kohr ldquoUnivalent mappings associated withthe Roper-Suffridge extension operatorrdquo Journal drsquoAnalyseMathematique vol 81 pp 331ndash342 2000

[16] S Gong and T S Liu ldquoThe generalized Roper-Suffridge exten-sion operatorrdquo Journal of Mathematical Analysis and Applica-tions vol 284 no 2 pp 425ndash434 2003

[17] G Kohr ldquoLoewner chains and a modification of the Roper-Suffridge extension operatorrdquo Mathematica vol 71 no 1 pp41ndash48 2006

[18] J R Muir ldquoA class of Loewner chain preserving extensionoperatorsrdquo Journal of Mathematical Analysis and Applicationsvol 337 no 2 pp 862ndash879 2008

[19] Y H Zhao Almost Starlike Mappings of Complex Order 120582 onthe Unit Ball of a Complex Banach Space Zhejiang NormalUniversity Jinhua China 2013

[20] L V Ahlfors Complex Analysis McGraw-Hill New York NYUSA 3rd edition 1978

[21] L Bieberbach ldquoUber einige extremal problem in gebiete derkonformen abbildungrdquo Mathematische Annalen vol 77 no 2pp 153ndash172 1916

[22] T Hayami S Owa and H M Srivastava ldquoCoefficient inequal-ities for certain classes of analytic and univalent functionsrdquoJournal of Inequalities in Pure and Applied Mathematics vol 8no 4 pp 1ndash10 2007

[23] M S Liu and Y C Zhu ldquoGeneralized Roper-Suffridge oper-ators on bounded and complete Reinhardt domainsrdquo Sciencein China vol 37 no 10 pp 1193ndash1206 2007

[24] Y Y Cui and C J Wang ldquoThe generalized Roper-Suffridgeextension operators in complex Banach spacerdquo Journal ofMathematics vol 34 no 3 pp 515ndash520 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Discrete Dynamics in Nature and Society

Remark 17 Setting 120582 = 120572(120572 minus 1) in Theorems 14ndash16 weget the corresponding results for almost starlike functions oforder 1205724 The Invariance of Almost Starlike Mappings

of Complex Order 120582In this section we mainly discuss the invariance of almoststarlike mappings of complex order 120582 on Reinhardt domainsand on the unit ball in complex Banach spaces under somegeneralized Roper-Suffridge operators

Lemma 18 (see [23]) Let Ω sub C119899 be a bounded andcompletely Reinhardt domain whose Minkowski functional120588(119911) is 1198621 except for a lower-dimensional manifold Ω0 Thenfor forall119911 = (1199111 119911119899) isin Ω Ω0 we have120597120588 (119911)120597119911 = (120597120588 (119911)1205971199111 120597120588 (119911)1205971199112 120597120588 (119911)120597119911119899 )

120597120588 (119911)120597119911119895 119911119895 ge 0120588 (119911) = 2 119899sum

119895=1

120597120588 (119911)120597119911119895 119911119895(65)

Theorem 19 Let 119891(119911) be an almost starlike function of com-plex order 120582 on 119863 with 120582 isin C and Re 120582 le 0 Let Ω sub C119899 be abounded and completely Reinhardt domain whose Minkowskifunctional 120588(119911) is 1198621 on Ω 0 Let

119865 (119911) = (119903119891(1199111119903 ) (119903119891 (1199111119903)1199111 )1205732sdot 1199112 (119903119891 (1199111119903)1199111 )120573119899 119911119899)1015840

(66)

where 120573119895 isin [0 1] 119903 = sup|1199111| 119911 = (1199111 119911119899)1015840 isin Ω and(119903119891(1199111119903)1199111)120573119895 |1199111=0 = 1 (119895 = 2 119899) Then 119865(119911) is an almoststarlike mapping of complex order 120582 on ΩProof For

119865 (119911) = (119903119891(1199111119903 ) (119903119891 (1199111119903)1199111 )1205732sdot 1199112 (119903119891 (1199111119903)1199111 )120573119899 119911119899)1015840

(67)

immediately we have

119869119865 (119911) =(((((((((((

1198911015840 (1199111119903 ) 0 sdot sdot sdot 01205732119903 [1198911015840 (1199111119903)119891 (1199111119903) minus 1199031199111 ](119903119891 (1199111119903)1199111 )1205732 1199112 (119903119891 (1199111119903)1199111 )1205732 sdot sdot sdot 0

sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot120573119899119903 [1198911015840 (1199111119903)119891 (1199111119903) minus 1199031199111](119903119891 (1199111119903)1199111 )120573119899 119911119899 0 sdot sdot sdot (119903119891 (1199111119903)1199111 )120573119899

)))))))))))

(68)

Therefore

119869minus1119865 (119911) =(((((((((((

11198911015840 (1199111119903) 0 sdot sdot sdot 01205732119903 [ 11990311991111198911015840 (1199111119903) minus 1119891 (1199111119903)] 1199112 (119903119891 (1199111119903)1199111 )minus1205732 sdot sdot sdot 0

sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot120573119899119903 [ 11990311991111198911015840 (1199111119903) minus 1119891 (1199111119903)] 119911119899 0 sdot sdot sdot (119903119891 (1199111119903)1199111 )minus120573119899

)))))))))))

(69)

Discrete Dynamics in Nature and Society 9

Then 119869minus1119865 (119911) 119865 (119911)= (119903119891 (1199111119903)1198911015840 (1199111119903) (1 minus 1205732 + 1205732 119903119891 (1199111119903)11991111198911015840 (1199111119903))sdot 1199112 (1 minus 120573119899 + 120573119899 119903119891 (1199111119903)11991111198911015840 (1199111119903)) 119911119899)1015840

(70)

By Lemma 18 we get(1 minus 120582) 2120597120588120597119911 (119911) 119869minus1119865 (119911) 119865 (119911) = 2 (1 minus 120582)sdot [[ 119903119891 (1199111119903)11991111198911015840 (1199111119903) 120597120588 (119911)1205971199111 1199111+ 119899sum119895=2

(1 minus 120573119895 + 120573119895 119903119891 (1199111119903)11991111198911015840 (1199111119903)) 120597120588 (119911)120597119911119895 119911119895]] (71)

Since119891(119911) is an almost starlike function of complex order120582 thenR[(1 minus 120582) 119903119891 (1199111119903)11991111198911015840 (1199111119903)] ge minusR120582 (72)

Therefore by Lemma 18 we obtain

R [(1 minus 120582) 2120597120588120597119911 (119911) 119869minus1119865 (119911) 119865 (119911)] = R[(1 minus 120582)sdot 119903119891 (1199111119903)11991111198911015840 (1199111119903)] 2120597120588 (119911)1205971199111 1199111 + 119899sum

119895=2

(1 minus 120573119895)R (1 minus 120582)+ 120573119895R[(1 minus 120582) 119903119891 (1199111119903)11991111198911015840 (1199111119903)] 2120597120588 (119911)120597119911119895 119911119895 ge minusR120582sdot 2120597120588 (119911)1205971199111 1199111 + 119899sum

119895=2

(1 minus 120573119895) (1 minusR120582) + 120573119895 (minusR120582)sdot 2120597120588 (119911)120597119911119895 119911119895 = (minusR120582) [2120597120588 (119911)1205971199111 1199111 + 2120597120588 (119911)120597119911119895 119911119895]+ 119899sum119895=2

(1 minus 120573119895) 2120597120588 (119911)120597119911119895 119911119895 ge (minusR120582) 120588 (119911)

(73)

Hence 119865(119911) is an almost starlike mapping of complex order 120582onΩ by Definition 4

Theorem 20 Let 119891119895 (119895 = 1 2 119899) be almost starlikefunctions of complex order 120582 on 119863 with 120582 isin C and Re 120582 lt 0Let Ω sub C119899 be a bounded starlike and circular domain whoseMinkowski functional 120588(119911) is1198621 onΩ0 Let 119903119895 be the radiusof the disk 119880119895 = 119911119895 isin C 119911 = (1199111 119911119895 119911119899)1015840 isin Ω whosecenter is zero Let

119865 (119911) = 119911 119899prod119895=1

(119903119895119891119895 (119911119895119903119895)119911119895 )120582119894 (74)

where 120582119894119895 ge 0 sum119899119895=1 120582119894119895 = 1 and (119891119895(119911119895)119911119895)120582119894119895 |119911119895=0 = 1 (119894 119895 =1 2 119899)Then119865(119911) is an almost starlikemapping of complexorder 120582 on ΩProof Let

120577119895 = 119911119895119903119895 119892 (119911) = 119899prod

119895=1

(119891119895 (120577119895)120577119895 )120582119894 (75)

which follows 119865(119911) = 119911119892(119911) Then 119869119865(119911) = 119869119892(119911)119911 + 119892(119911) Bydirect computation we get

119869119892 (119911) 119911 = 119892 (119911) [[119899sum119895=1

1205821198951198911015840119895 (119911119895119903119895) 119911119895119903119895119891119895 (119911119895119903119895) minus 1]]= 119892 (119911) [[

119899sum119895=1

1205821198951198911015840119895 (120577119895) 120577119895119891119895 (120577119895) minus 1]] (76)

Since 119891119895 are almost starlike functions of complex order 120582 on119863 then

R[[(1 minus 120582) 119891119895 (120577119895)1205771198951198911015840119895 (120577119895)]] ge minusR120582 gt 0 (77)

Therefore

R[ 11 minus 120582 (1 + 119869119892 (119911) 119911119892 (119911) )]= R[[ 11 minus 120582 119899sum

119895=1

1205821198951198911015840119895 (120577119895) 120577119895119891119895 (120577119895) ]]= 119899sum119895=1

120582119895R[[ 11 minus 120582 1198911015840119895 (120577119895) 120577119895119891119895 (120577119895) ]] gt 0(78)

Then 119869119892(119911)119911+119892(119911) = 0 It is obvious that119865(0) = 0 and 119869119865(0) =119868 so 119865(119911) is a normalized locally biholomorphic mapping onΩ and

119869minus1119865 (119911) 119865 (119911) = 119892 (119911) 119911119869119892 (119911) 119911 + 119892 (119911) (79)

Let

119902119895 (120577119895) = 2Re (minus120582) 1198911015840119895 (120577119895) 120577119895(1 minus 120582) 119891119895 (120577119895) minus 1 (80)

(77) follows 100381610038161003816100381610038161003816100381610038161003816100381610038162Re (minus120582) 1198911015840119895 (120577119895) 120577119895(1 minus 120582) 119891119895 (120577119895) minus 110038161003816100381610038161003816100381610038161003816100381610038161003816 le 1 (81)

10 Discrete Dynamics in Nature and Society

that is |119902119895(120577119895)| le 1 For Re 120582 lt 0 by (76) (79) and Lemma 18we obtain1003816100381610038161003816100381610038161003816100381610038162R (minus120582) 1(1 minus 120582) (2120588 (119911)) (120597120588120597119911) (119911) 119869minus1119865 (119911) 119865 (119911)

minus 1100381610038161003816100381610038161003816100381610038161003816 =100381610038161003816100381610038161003816100381610038161003816100381610038162R (minus120582) 11 minus 120582 119899sum

119895=1

1205821198951198911015840119895 (120577119895) 120577119895119891119895 (120577119895) minus 110038161003816100381610038161003816100381610038161003816100381610038161003816= 10038161003816100381610038161003816100381610038161003816100381610038161003816 119899sum119895=1120582119895119902119895 (120577119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816 le 119899sum119895=1120582119895 = 1(82)

which leads to

R [(1 minus 120582) 2120597120588120597119911 (119911) 119869minus1119865 (119911) 119865 (119911)] ge minusR120582120588 (119911) (83)

Hence 119865(119911) is an almost starlike mapping of complex order 120582onΩ by Definition 4

Similar to Theorem 20 we can get the following conclu-sion

Theorem 21 Let 119891119895 (119895 = 1 2 119899) be almost starlikefunctions of complex order 120582 on 119863 with 120582 isin C and Re 120582 lt 0Let

119865 (119909) = 119909 119899prod119895=1

(119891119895 (119879119906119895 (119909))119879119906119895 (119909) )120582119895 119909 isin 119861 (84)

where 120582119894119895 ge 0 sum119899119895=1 120582119894119895 = 1 (119891119895(119909119895)119909119895)120582119894119895 |119909119895=0 = 1 (119909119895 =119879119906119895(119909) 119894 119895 = 1 2 119899) and 119906119895 is the unit vector of thecomplex Banach space 119883 Then 119865(119909) is an almost starlikemapping of complex order 120582 on the unit ball 119861 in complexBanach space119883

Theorem 22 Let 119891 be an almost starlike function of complexorder 120582 on119863 with 120582 isin C and Re 120582 le 0 Let

119865 (119909) = 119899minus1sum119895=1

(119891(1198791199091 (119909))1198791199091 (119909) )120573119895 119879119909119895 (119909) 119909119895 + 119909minus 119899minus1sum119895=1

119879119909119895 (119909) 119909119895 119909 isin 119861 (85)

where 1199091 isin 119861 1199091 = 1 120573119895 isin [0 1] (119891(119911)119911)120573119895 |119911=0 = 1 (119895 =1 2 119899 minus 1) and 1199091 119909119899 are linearly independent and119879119909119894 isin 119883lowast such that 119879119909119894 = 1 119879119909119894(119909119894) = 1 and 119879119909119894(119909119895) =0 (119894 = 119895) for forall119909119894 Then 119865(119909) is an almost starlike mapping ofcomplex order 120582 on the unit ball 119861 in complex Banach space119883

Proof From [24] we get 119865(119909) is a normalized biholomorphicmapping on 119861 and119909 119879119909 [119863119865 (119909)minus1 119865 (119909)]= 1199092

+ 119899minus1sum119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162 [ 119891 (1198791199091 (119909))1198791199091 (119909) 1198911015840 (1198791199091 (119909)) minus 1] (86)

Since119891 is an almost starlike function of complex order120582 thenR[[(1 minus 120582) 119891 (119879119909119895 (119909))119879119909119895 (119909) 1198911015840 (119879119909119895 (119909))]] ge minusR120582 (87)

Therefore

R (1 minus 120582) 119879119909 [119863119865 (119909)minus1 119865 (119909)] = (1 minusR120582) 119909+ 119899minus1sum119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162119909sdotR(1 minus 120582) [ 119891 (1198791199091 (119909))1198791199091 (119909) 1198911015840 (1198791199091 (119909)) minus 1]ge (1 minusR120582) 119909 + 119899minus1sum

119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162119909sdot [R (minus120582) minus (1 minusR120582)] = minusR120582 119909+ 1119909 [[1199092 minus

119899minus1sum119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162]] ge minusR120582 119909

(88)

which lead to the desired conclusion by Definition 5

Similar to Theorem 22 we can get the following conclu-sion

Theorem 23 Let 119891 be an almost starlike function of complexorder 120582 on119863 with 120582 isin C Re 120582 le 0 Let119865 (119909) = Φ120573 (119891) (119909)= 119891 (1198791199091 (119909)) 1199091

+ (119891 (1198791199091 (119909))1198791199091 (119909) )120573 (119909 minus 1198791199091 (119909) 1199091) (89)

where 0 le 120573 le 1 1199091 isin 119861 1199091 = 1 (119891(1199111)1199111)120573|1199111=0 = 1 and119891(1199111) = 0 for 1199111 = 0 Then 119865(119909) is an almost starlike mappingof complex order 120582 on 119861Remark 24 Setting 120582 = 120572(120572 minus 1) in Theorems 19ndash23 weget the corresponding results with respect to almost starlikemappings of order 120572

Discrete Dynamics in Nature and Society 11

Competing Interests

The authors declare that they have no conflict of interests

Acknowledgments

This work is supported by NSF of China (no U1204618) andScience and Technology Research Projects of Henan Provin-cial Education Department (nos 14B110015 and 14B110016)

References

[1] C Pommerenke Univalent Functions Vandenhoeck amp Ru-precht Gottingen Germany 1975

[2] H Cartan ldquoSur la possibilite drsquoentendre aux functions de plus-ieurs variables complexes la theorie des functions univalentsrdquoin Lecons sur les Functions Univalents onMutivalents P MontelEd Gauthier-Villar Paris France 1933

[3] R W Barnard C H FitzGerald and S Gong ldquoA distortionthoerem for biholomorphicmappings inC2rdquoTransactions of theAmerican Mathematical Society vol 344 pp 902ndash924 1994

[4] T S Liu andW J Zhang ldquoA distortion thoerem for biholomor-phic convex mappings in C119899rdquo Chinese Journal of ContemporaryMathematics vol 20 no 3 pp 421ndash430 1999

[5] S Gong S K Wang and Q H Yu ldquoBiholomorphic convexmappings of ball in C119899rdquo Pacific Journal of Mathematics vol 161no 2 pp 287ndash306 1993

[6] S Gong and T S Liu ldquoDistortion theorems for biholomorphicconvex mappings on bounded convex circular domainsrdquo Chi-nese Annals of Mathematics Series B vol 20 no 3 pp 297ndash3041999

[7] I Graham and D Varolin ldquoBloch constants in one and severalvariablesrdquoPacific Journal ofMathematics vol 174 no 2 pp 347ndash357 1996

[8] P L DurenUnivalent Functions Springer New York NY USA1983

[9] T H Gronwall ldquoSome remarks on conformal representationrdquoAnnals of Mathematics vol 16 no 1ndash4 pp 72ndash76 191415

[10] M I Robertson ldquoOn the theory of univalent functionsrdquo Annalsof Mathematics vol 37 no 2 pp 374ndash408 1936

[11] A V Boyd ldquoCoefficient estimates for starlike functions of order120572rdquo Proceedings of the AmericanMathematical Society vol 17 pp1016ndash1018 1966

[12] H Silverman ldquoUnivalent functions with negative coefficientsrdquoProceedings of the American Mathematical Society vol 51 pp109ndash116 1975

[13] H M Srivastava and S Owa ldquoCertain subclasses of starlikefunctions Irdquo Journal ofMathematical Analysis and Applicationsvol 161 no 2 pp 405ndash415 1991

[14] K A Roper and T J Suffridge ldquoConvex mappings on the unitball ofC119899rdquo Journal drsquoAnalyseMathematique vol 65 pp 333ndash3471995

[15] I Graham and G Kohr ldquoUnivalent mappings associated withthe Roper-Suffridge extension operatorrdquo Journal drsquoAnalyseMathematique vol 81 pp 331ndash342 2000

[16] S Gong and T S Liu ldquoThe generalized Roper-Suffridge exten-sion operatorrdquo Journal of Mathematical Analysis and Applica-tions vol 284 no 2 pp 425ndash434 2003

[17] G Kohr ldquoLoewner chains and a modification of the Roper-Suffridge extension operatorrdquo Mathematica vol 71 no 1 pp41ndash48 2006

[18] J R Muir ldquoA class of Loewner chain preserving extensionoperatorsrdquo Journal of Mathematical Analysis and Applicationsvol 337 no 2 pp 862ndash879 2008

[19] Y H Zhao Almost Starlike Mappings of Complex Order 120582 onthe Unit Ball of a Complex Banach Space Zhejiang NormalUniversity Jinhua China 2013

[20] L V Ahlfors Complex Analysis McGraw-Hill New York NYUSA 3rd edition 1978

[21] L Bieberbach ldquoUber einige extremal problem in gebiete derkonformen abbildungrdquo Mathematische Annalen vol 77 no 2pp 153ndash172 1916

[22] T Hayami S Owa and H M Srivastava ldquoCoefficient inequal-ities for certain classes of analytic and univalent functionsrdquoJournal of Inequalities in Pure and Applied Mathematics vol 8no 4 pp 1ndash10 2007

[23] M S Liu and Y C Zhu ldquoGeneralized Roper-Suffridge oper-ators on bounded and complete Reinhardt domainsrdquo Sciencein China vol 37 no 10 pp 1193ndash1206 2007

[24] Y Y Cui and C J Wang ldquoThe generalized Roper-Suffridgeextension operators in complex Banach spacerdquo Journal ofMathematics vol 34 no 3 pp 515ndash520 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 9

Then 119869minus1119865 (119911) 119865 (119911)= (119903119891 (1199111119903)1198911015840 (1199111119903) (1 minus 1205732 + 1205732 119903119891 (1199111119903)11991111198911015840 (1199111119903))sdot 1199112 (1 minus 120573119899 + 120573119899 119903119891 (1199111119903)11991111198911015840 (1199111119903)) 119911119899)1015840

(70)

By Lemma 18 we get(1 minus 120582) 2120597120588120597119911 (119911) 119869minus1119865 (119911) 119865 (119911) = 2 (1 minus 120582)sdot [[ 119903119891 (1199111119903)11991111198911015840 (1199111119903) 120597120588 (119911)1205971199111 1199111+ 119899sum119895=2

(1 minus 120573119895 + 120573119895 119903119891 (1199111119903)11991111198911015840 (1199111119903)) 120597120588 (119911)120597119911119895 119911119895]] (71)

Since119891(119911) is an almost starlike function of complex order120582 thenR[(1 minus 120582) 119903119891 (1199111119903)11991111198911015840 (1199111119903)] ge minusR120582 (72)

Therefore by Lemma 18 we obtain

R [(1 minus 120582) 2120597120588120597119911 (119911) 119869minus1119865 (119911) 119865 (119911)] = R[(1 minus 120582)sdot 119903119891 (1199111119903)11991111198911015840 (1199111119903)] 2120597120588 (119911)1205971199111 1199111 + 119899sum

119895=2

(1 minus 120573119895)R (1 minus 120582)+ 120573119895R[(1 minus 120582) 119903119891 (1199111119903)11991111198911015840 (1199111119903)] 2120597120588 (119911)120597119911119895 119911119895 ge minusR120582sdot 2120597120588 (119911)1205971199111 1199111 + 119899sum

119895=2

(1 minus 120573119895) (1 minusR120582) + 120573119895 (minusR120582)sdot 2120597120588 (119911)120597119911119895 119911119895 = (minusR120582) [2120597120588 (119911)1205971199111 1199111 + 2120597120588 (119911)120597119911119895 119911119895]+ 119899sum119895=2

(1 minus 120573119895) 2120597120588 (119911)120597119911119895 119911119895 ge (minusR120582) 120588 (119911)

(73)

Hence 119865(119911) is an almost starlike mapping of complex order 120582onΩ by Definition 4

Theorem 20 Let 119891119895 (119895 = 1 2 119899) be almost starlikefunctions of complex order 120582 on 119863 with 120582 isin C and Re 120582 lt 0Let Ω sub C119899 be a bounded starlike and circular domain whoseMinkowski functional 120588(119911) is1198621 onΩ0 Let 119903119895 be the radiusof the disk 119880119895 = 119911119895 isin C 119911 = (1199111 119911119895 119911119899)1015840 isin Ω whosecenter is zero Let

119865 (119911) = 119911 119899prod119895=1

(119903119895119891119895 (119911119895119903119895)119911119895 )120582119894 (74)

where 120582119894119895 ge 0 sum119899119895=1 120582119894119895 = 1 and (119891119895(119911119895)119911119895)120582119894119895 |119911119895=0 = 1 (119894 119895 =1 2 119899)Then119865(119911) is an almost starlikemapping of complexorder 120582 on ΩProof Let

120577119895 = 119911119895119903119895 119892 (119911) = 119899prod

119895=1

(119891119895 (120577119895)120577119895 )120582119894 (75)

which follows 119865(119911) = 119911119892(119911) Then 119869119865(119911) = 119869119892(119911)119911 + 119892(119911) Bydirect computation we get

119869119892 (119911) 119911 = 119892 (119911) [[119899sum119895=1

1205821198951198911015840119895 (119911119895119903119895) 119911119895119903119895119891119895 (119911119895119903119895) minus 1]]= 119892 (119911) [[

119899sum119895=1

1205821198951198911015840119895 (120577119895) 120577119895119891119895 (120577119895) minus 1]] (76)

Since 119891119895 are almost starlike functions of complex order 120582 on119863 then

R[[(1 minus 120582) 119891119895 (120577119895)1205771198951198911015840119895 (120577119895)]] ge minusR120582 gt 0 (77)

Therefore

R[ 11 minus 120582 (1 + 119869119892 (119911) 119911119892 (119911) )]= R[[ 11 minus 120582 119899sum

119895=1

1205821198951198911015840119895 (120577119895) 120577119895119891119895 (120577119895) ]]= 119899sum119895=1

120582119895R[[ 11 minus 120582 1198911015840119895 (120577119895) 120577119895119891119895 (120577119895) ]] gt 0(78)

Then 119869119892(119911)119911+119892(119911) = 0 It is obvious that119865(0) = 0 and 119869119865(0) =119868 so 119865(119911) is a normalized locally biholomorphic mapping onΩ and

119869minus1119865 (119911) 119865 (119911) = 119892 (119911) 119911119869119892 (119911) 119911 + 119892 (119911) (79)

Let

119902119895 (120577119895) = 2Re (minus120582) 1198911015840119895 (120577119895) 120577119895(1 minus 120582) 119891119895 (120577119895) minus 1 (80)

(77) follows 100381610038161003816100381610038161003816100381610038161003816100381610038162Re (minus120582) 1198911015840119895 (120577119895) 120577119895(1 minus 120582) 119891119895 (120577119895) minus 110038161003816100381610038161003816100381610038161003816100381610038161003816 le 1 (81)

10 Discrete Dynamics in Nature and Society

that is |119902119895(120577119895)| le 1 For Re 120582 lt 0 by (76) (79) and Lemma 18we obtain1003816100381610038161003816100381610038161003816100381610038162R (minus120582) 1(1 minus 120582) (2120588 (119911)) (120597120588120597119911) (119911) 119869minus1119865 (119911) 119865 (119911)

minus 1100381610038161003816100381610038161003816100381610038161003816 =100381610038161003816100381610038161003816100381610038161003816100381610038162R (minus120582) 11 minus 120582 119899sum

119895=1

1205821198951198911015840119895 (120577119895) 120577119895119891119895 (120577119895) minus 110038161003816100381610038161003816100381610038161003816100381610038161003816= 10038161003816100381610038161003816100381610038161003816100381610038161003816 119899sum119895=1120582119895119902119895 (120577119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816 le 119899sum119895=1120582119895 = 1(82)

which leads to

R [(1 minus 120582) 2120597120588120597119911 (119911) 119869minus1119865 (119911) 119865 (119911)] ge minusR120582120588 (119911) (83)

Hence 119865(119911) is an almost starlike mapping of complex order 120582onΩ by Definition 4

Similar to Theorem 20 we can get the following conclu-sion

Theorem 21 Let 119891119895 (119895 = 1 2 119899) be almost starlikefunctions of complex order 120582 on 119863 with 120582 isin C and Re 120582 lt 0Let

119865 (119909) = 119909 119899prod119895=1

(119891119895 (119879119906119895 (119909))119879119906119895 (119909) )120582119895 119909 isin 119861 (84)

where 120582119894119895 ge 0 sum119899119895=1 120582119894119895 = 1 (119891119895(119909119895)119909119895)120582119894119895 |119909119895=0 = 1 (119909119895 =119879119906119895(119909) 119894 119895 = 1 2 119899) and 119906119895 is the unit vector of thecomplex Banach space 119883 Then 119865(119909) is an almost starlikemapping of complex order 120582 on the unit ball 119861 in complexBanach space119883

Theorem 22 Let 119891 be an almost starlike function of complexorder 120582 on119863 with 120582 isin C and Re 120582 le 0 Let

119865 (119909) = 119899minus1sum119895=1

(119891(1198791199091 (119909))1198791199091 (119909) )120573119895 119879119909119895 (119909) 119909119895 + 119909minus 119899minus1sum119895=1

119879119909119895 (119909) 119909119895 119909 isin 119861 (85)

where 1199091 isin 119861 1199091 = 1 120573119895 isin [0 1] (119891(119911)119911)120573119895 |119911=0 = 1 (119895 =1 2 119899 minus 1) and 1199091 119909119899 are linearly independent and119879119909119894 isin 119883lowast such that 119879119909119894 = 1 119879119909119894(119909119894) = 1 and 119879119909119894(119909119895) =0 (119894 = 119895) for forall119909119894 Then 119865(119909) is an almost starlike mapping ofcomplex order 120582 on the unit ball 119861 in complex Banach space119883

Proof From [24] we get 119865(119909) is a normalized biholomorphicmapping on 119861 and119909 119879119909 [119863119865 (119909)minus1 119865 (119909)]= 1199092

+ 119899minus1sum119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162 [ 119891 (1198791199091 (119909))1198791199091 (119909) 1198911015840 (1198791199091 (119909)) minus 1] (86)

Since119891 is an almost starlike function of complex order120582 thenR[[(1 minus 120582) 119891 (119879119909119895 (119909))119879119909119895 (119909) 1198911015840 (119879119909119895 (119909))]] ge minusR120582 (87)

Therefore

R (1 minus 120582) 119879119909 [119863119865 (119909)minus1 119865 (119909)] = (1 minusR120582) 119909+ 119899minus1sum119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162119909sdotR(1 minus 120582) [ 119891 (1198791199091 (119909))1198791199091 (119909) 1198911015840 (1198791199091 (119909)) minus 1]ge (1 minusR120582) 119909 + 119899minus1sum

119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162119909sdot [R (minus120582) minus (1 minusR120582)] = minusR120582 119909+ 1119909 [[1199092 minus

119899minus1sum119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162]] ge minusR120582 119909

(88)

which lead to the desired conclusion by Definition 5

Similar to Theorem 22 we can get the following conclu-sion

Theorem 23 Let 119891 be an almost starlike function of complexorder 120582 on119863 with 120582 isin C Re 120582 le 0 Let119865 (119909) = Φ120573 (119891) (119909)= 119891 (1198791199091 (119909)) 1199091

+ (119891 (1198791199091 (119909))1198791199091 (119909) )120573 (119909 minus 1198791199091 (119909) 1199091) (89)

where 0 le 120573 le 1 1199091 isin 119861 1199091 = 1 (119891(1199111)1199111)120573|1199111=0 = 1 and119891(1199111) = 0 for 1199111 = 0 Then 119865(119909) is an almost starlike mappingof complex order 120582 on 119861Remark 24 Setting 120582 = 120572(120572 minus 1) in Theorems 19ndash23 weget the corresponding results with respect to almost starlikemappings of order 120572

Discrete Dynamics in Nature and Society 11

Competing Interests

The authors declare that they have no conflict of interests

Acknowledgments

This work is supported by NSF of China (no U1204618) andScience and Technology Research Projects of Henan Provin-cial Education Department (nos 14B110015 and 14B110016)

References

[1] C Pommerenke Univalent Functions Vandenhoeck amp Ru-precht Gottingen Germany 1975

[2] H Cartan ldquoSur la possibilite drsquoentendre aux functions de plus-ieurs variables complexes la theorie des functions univalentsrdquoin Lecons sur les Functions Univalents onMutivalents P MontelEd Gauthier-Villar Paris France 1933

[3] R W Barnard C H FitzGerald and S Gong ldquoA distortionthoerem for biholomorphicmappings inC2rdquoTransactions of theAmerican Mathematical Society vol 344 pp 902ndash924 1994

[4] T S Liu andW J Zhang ldquoA distortion thoerem for biholomor-phic convex mappings in C119899rdquo Chinese Journal of ContemporaryMathematics vol 20 no 3 pp 421ndash430 1999

[5] S Gong S K Wang and Q H Yu ldquoBiholomorphic convexmappings of ball in C119899rdquo Pacific Journal of Mathematics vol 161no 2 pp 287ndash306 1993

[6] S Gong and T S Liu ldquoDistortion theorems for biholomorphicconvex mappings on bounded convex circular domainsrdquo Chi-nese Annals of Mathematics Series B vol 20 no 3 pp 297ndash3041999

[7] I Graham and D Varolin ldquoBloch constants in one and severalvariablesrdquoPacific Journal ofMathematics vol 174 no 2 pp 347ndash357 1996

[8] P L DurenUnivalent Functions Springer New York NY USA1983

[9] T H Gronwall ldquoSome remarks on conformal representationrdquoAnnals of Mathematics vol 16 no 1ndash4 pp 72ndash76 191415

[10] M I Robertson ldquoOn the theory of univalent functionsrdquo Annalsof Mathematics vol 37 no 2 pp 374ndash408 1936

[11] A V Boyd ldquoCoefficient estimates for starlike functions of order120572rdquo Proceedings of the AmericanMathematical Society vol 17 pp1016ndash1018 1966

[12] H Silverman ldquoUnivalent functions with negative coefficientsrdquoProceedings of the American Mathematical Society vol 51 pp109ndash116 1975

[13] H M Srivastava and S Owa ldquoCertain subclasses of starlikefunctions Irdquo Journal ofMathematical Analysis and Applicationsvol 161 no 2 pp 405ndash415 1991

[14] K A Roper and T J Suffridge ldquoConvex mappings on the unitball ofC119899rdquo Journal drsquoAnalyseMathematique vol 65 pp 333ndash3471995

[15] I Graham and G Kohr ldquoUnivalent mappings associated withthe Roper-Suffridge extension operatorrdquo Journal drsquoAnalyseMathematique vol 81 pp 331ndash342 2000

[16] S Gong and T S Liu ldquoThe generalized Roper-Suffridge exten-sion operatorrdquo Journal of Mathematical Analysis and Applica-tions vol 284 no 2 pp 425ndash434 2003

[17] G Kohr ldquoLoewner chains and a modification of the Roper-Suffridge extension operatorrdquo Mathematica vol 71 no 1 pp41ndash48 2006

[18] J R Muir ldquoA class of Loewner chain preserving extensionoperatorsrdquo Journal of Mathematical Analysis and Applicationsvol 337 no 2 pp 862ndash879 2008

[19] Y H Zhao Almost Starlike Mappings of Complex Order 120582 onthe Unit Ball of a Complex Banach Space Zhejiang NormalUniversity Jinhua China 2013

[20] L V Ahlfors Complex Analysis McGraw-Hill New York NYUSA 3rd edition 1978

[21] L Bieberbach ldquoUber einige extremal problem in gebiete derkonformen abbildungrdquo Mathematische Annalen vol 77 no 2pp 153ndash172 1916

[22] T Hayami S Owa and H M Srivastava ldquoCoefficient inequal-ities for certain classes of analytic and univalent functionsrdquoJournal of Inequalities in Pure and Applied Mathematics vol 8no 4 pp 1ndash10 2007

[23] M S Liu and Y C Zhu ldquoGeneralized Roper-Suffridge oper-ators on bounded and complete Reinhardt domainsrdquo Sciencein China vol 37 no 10 pp 1193ndash1206 2007

[24] Y Y Cui and C J Wang ldquoThe generalized Roper-Suffridgeextension operators in complex Banach spacerdquo Journal ofMathematics vol 34 no 3 pp 515ndash520 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Discrete Dynamics in Nature and Society

that is |119902119895(120577119895)| le 1 For Re 120582 lt 0 by (76) (79) and Lemma 18we obtain1003816100381610038161003816100381610038161003816100381610038162R (minus120582) 1(1 minus 120582) (2120588 (119911)) (120597120588120597119911) (119911) 119869minus1119865 (119911) 119865 (119911)

minus 1100381610038161003816100381610038161003816100381610038161003816 =100381610038161003816100381610038161003816100381610038161003816100381610038162R (minus120582) 11 minus 120582 119899sum

119895=1

1205821198951198911015840119895 (120577119895) 120577119895119891119895 (120577119895) minus 110038161003816100381610038161003816100381610038161003816100381610038161003816= 10038161003816100381610038161003816100381610038161003816100381610038161003816 119899sum119895=1120582119895119902119895 (120577119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816 le 119899sum119895=1120582119895 = 1(82)

which leads to

R [(1 minus 120582) 2120597120588120597119911 (119911) 119869minus1119865 (119911) 119865 (119911)] ge minusR120582120588 (119911) (83)

Hence 119865(119911) is an almost starlike mapping of complex order 120582onΩ by Definition 4

Similar to Theorem 20 we can get the following conclu-sion

Theorem 21 Let 119891119895 (119895 = 1 2 119899) be almost starlikefunctions of complex order 120582 on 119863 with 120582 isin C and Re 120582 lt 0Let

119865 (119909) = 119909 119899prod119895=1

(119891119895 (119879119906119895 (119909))119879119906119895 (119909) )120582119895 119909 isin 119861 (84)

where 120582119894119895 ge 0 sum119899119895=1 120582119894119895 = 1 (119891119895(119909119895)119909119895)120582119894119895 |119909119895=0 = 1 (119909119895 =119879119906119895(119909) 119894 119895 = 1 2 119899) and 119906119895 is the unit vector of thecomplex Banach space 119883 Then 119865(119909) is an almost starlikemapping of complex order 120582 on the unit ball 119861 in complexBanach space119883

Theorem 22 Let 119891 be an almost starlike function of complexorder 120582 on119863 with 120582 isin C and Re 120582 le 0 Let

119865 (119909) = 119899minus1sum119895=1

(119891(1198791199091 (119909))1198791199091 (119909) )120573119895 119879119909119895 (119909) 119909119895 + 119909minus 119899minus1sum119895=1

119879119909119895 (119909) 119909119895 119909 isin 119861 (85)

where 1199091 isin 119861 1199091 = 1 120573119895 isin [0 1] (119891(119911)119911)120573119895 |119911=0 = 1 (119895 =1 2 119899 minus 1) and 1199091 119909119899 are linearly independent and119879119909119894 isin 119883lowast such that 119879119909119894 = 1 119879119909119894(119909119894) = 1 and 119879119909119894(119909119895) =0 (119894 = 119895) for forall119909119894 Then 119865(119909) is an almost starlike mapping ofcomplex order 120582 on the unit ball 119861 in complex Banach space119883

Proof From [24] we get 119865(119909) is a normalized biholomorphicmapping on 119861 and119909 119879119909 [119863119865 (119909)minus1 119865 (119909)]= 1199092

+ 119899minus1sum119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162 [ 119891 (1198791199091 (119909))1198791199091 (119909) 1198911015840 (1198791199091 (119909)) minus 1] (86)

Since119891 is an almost starlike function of complex order120582 thenR[[(1 minus 120582) 119891 (119879119909119895 (119909))119879119909119895 (119909) 1198911015840 (119879119909119895 (119909))]] ge minusR120582 (87)

Therefore

R (1 minus 120582) 119879119909 [119863119865 (119909)minus1 119865 (119909)] = (1 minusR120582) 119909+ 119899minus1sum119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162119909sdotR(1 minus 120582) [ 119891 (1198791199091 (119909))1198791199091 (119909) 1198911015840 (1198791199091 (119909)) minus 1]ge (1 minusR120582) 119909 + 119899minus1sum

119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162119909sdot [R (minus120582) minus (1 minusR120582)] = minusR120582 119909+ 1119909 [[1199092 minus

119899minus1sum119895=1

120573119895 100381610038161003816100381610038161003816119879119909119895 (119909)1003816100381610038161003816100381610038162]] ge minusR120582 119909

(88)

which lead to the desired conclusion by Definition 5

Similar to Theorem 22 we can get the following conclu-sion

Theorem 23 Let 119891 be an almost starlike function of complexorder 120582 on119863 with 120582 isin C Re 120582 le 0 Let119865 (119909) = Φ120573 (119891) (119909)= 119891 (1198791199091 (119909)) 1199091

+ (119891 (1198791199091 (119909))1198791199091 (119909) )120573 (119909 minus 1198791199091 (119909) 1199091) (89)

where 0 le 120573 le 1 1199091 isin 119861 1199091 = 1 (119891(1199111)1199111)120573|1199111=0 = 1 and119891(1199111) = 0 for 1199111 = 0 Then 119865(119909) is an almost starlike mappingof complex order 120582 on 119861Remark 24 Setting 120582 = 120572(120572 minus 1) in Theorems 19ndash23 weget the corresponding results with respect to almost starlikemappings of order 120572

Discrete Dynamics in Nature and Society 11

Competing Interests

The authors declare that they have no conflict of interests

Acknowledgments

This work is supported by NSF of China (no U1204618) andScience and Technology Research Projects of Henan Provin-cial Education Department (nos 14B110015 and 14B110016)

References

[1] C Pommerenke Univalent Functions Vandenhoeck amp Ru-precht Gottingen Germany 1975

[2] H Cartan ldquoSur la possibilite drsquoentendre aux functions de plus-ieurs variables complexes la theorie des functions univalentsrdquoin Lecons sur les Functions Univalents onMutivalents P MontelEd Gauthier-Villar Paris France 1933

[3] R W Barnard C H FitzGerald and S Gong ldquoA distortionthoerem for biholomorphicmappings inC2rdquoTransactions of theAmerican Mathematical Society vol 344 pp 902ndash924 1994

[4] T S Liu andW J Zhang ldquoA distortion thoerem for biholomor-phic convex mappings in C119899rdquo Chinese Journal of ContemporaryMathematics vol 20 no 3 pp 421ndash430 1999

[5] S Gong S K Wang and Q H Yu ldquoBiholomorphic convexmappings of ball in C119899rdquo Pacific Journal of Mathematics vol 161no 2 pp 287ndash306 1993

[6] S Gong and T S Liu ldquoDistortion theorems for biholomorphicconvex mappings on bounded convex circular domainsrdquo Chi-nese Annals of Mathematics Series B vol 20 no 3 pp 297ndash3041999

[7] I Graham and D Varolin ldquoBloch constants in one and severalvariablesrdquoPacific Journal ofMathematics vol 174 no 2 pp 347ndash357 1996

[8] P L DurenUnivalent Functions Springer New York NY USA1983

[9] T H Gronwall ldquoSome remarks on conformal representationrdquoAnnals of Mathematics vol 16 no 1ndash4 pp 72ndash76 191415

[10] M I Robertson ldquoOn the theory of univalent functionsrdquo Annalsof Mathematics vol 37 no 2 pp 374ndash408 1936

[11] A V Boyd ldquoCoefficient estimates for starlike functions of order120572rdquo Proceedings of the AmericanMathematical Society vol 17 pp1016ndash1018 1966

[12] H Silverman ldquoUnivalent functions with negative coefficientsrdquoProceedings of the American Mathematical Society vol 51 pp109ndash116 1975

[13] H M Srivastava and S Owa ldquoCertain subclasses of starlikefunctions Irdquo Journal ofMathematical Analysis and Applicationsvol 161 no 2 pp 405ndash415 1991

[14] K A Roper and T J Suffridge ldquoConvex mappings on the unitball ofC119899rdquo Journal drsquoAnalyseMathematique vol 65 pp 333ndash3471995

[15] I Graham and G Kohr ldquoUnivalent mappings associated withthe Roper-Suffridge extension operatorrdquo Journal drsquoAnalyseMathematique vol 81 pp 331ndash342 2000

[16] S Gong and T S Liu ldquoThe generalized Roper-Suffridge exten-sion operatorrdquo Journal of Mathematical Analysis and Applica-tions vol 284 no 2 pp 425ndash434 2003

[17] G Kohr ldquoLoewner chains and a modification of the Roper-Suffridge extension operatorrdquo Mathematica vol 71 no 1 pp41ndash48 2006

[18] J R Muir ldquoA class of Loewner chain preserving extensionoperatorsrdquo Journal of Mathematical Analysis and Applicationsvol 337 no 2 pp 862ndash879 2008

[19] Y H Zhao Almost Starlike Mappings of Complex Order 120582 onthe Unit Ball of a Complex Banach Space Zhejiang NormalUniversity Jinhua China 2013

[20] L V Ahlfors Complex Analysis McGraw-Hill New York NYUSA 3rd edition 1978

[21] L Bieberbach ldquoUber einige extremal problem in gebiete derkonformen abbildungrdquo Mathematische Annalen vol 77 no 2pp 153ndash172 1916

[22] T Hayami S Owa and H M Srivastava ldquoCoefficient inequal-ities for certain classes of analytic and univalent functionsrdquoJournal of Inequalities in Pure and Applied Mathematics vol 8no 4 pp 1ndash10 2007

[23] M S Liu and Y C Zhu ldquoGeneralized Roper-Suffridge oper-ators on bounded and complete Reinhardt domainsrdquo Sciencein China vol 37 no 10 pp 1193ndash1206 2007

[24] Y Y Cui and C J Wang ldquoThe generalized Roper-Suffridgeextension operators in complex Banach spacerdquo Journal ofMathematics vol 34 no 3 pp 515ndash520 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 11

Competing Interests

The authors declare that they have no conflict of interests

Acknowledgments

This work is supported by NSF of China (no U1204618) andScience and Technology Research Projects of Henan Provin-cial Education Department (nos 14B110015 and 14B110016)

References

[1] C Pommerenke Univalent Functions Vandenhoeck amp Ru-precht Gottingen Germany 1975

[2] H Cartan ldquoSur la possibilite drsquoentendre aux functions de plus-ieurs variables complexes la theorie des functions univalentsrdquoin Lecons sur les Functions Univalents onMutivalents P MontelEd Gauthier-Villar Paris France 1933

[3] R W Barnard C H FitzGerald and S Gong ldquoA distortionthoerem for biholomorphicmappings inC2rdquoTransactions of theAmerican Mathematical Society vol 344 pp 902ndash924 1994

[4] T S Liu andW J Zhang ldquoA distortion thoerem for biholomor-phic convex mappings in C119899rdquo Chinese Journal of ContemporaryMathematics vol 20 no 3 pp 421ndash430 1999

[5] S Gong S K Wang and Q H Yu ldquoBiholomorphic convexmappings of ball in C119899rdquo Pacific Journal of Mathematics vol 161no 2 pp 287ndash306 1993

[6] S Gong and T S Liu ldquoDistortion theorems for biholomorphicconvex mappings on bounded convex circular domainsrdquo Chi-nese Annals of Mathematics Series B vol 20 no 3 pp 297ndash3041999

[7] I Graham and D Varolin ldquoBloch constants in one and severalvariablesrdquoPacific Journal ofMathematics vol 174 no 2 pp 347ndash357 1996

[8] P L DurenUnivalent Functions Springer New York NY USA1983

[9] T H Gronwall ldquoSome remarks on conformal representationrdquoAnnals of Mathematics vol 16 no 1ndash4 pp 72ndash76 191415

[10] M I Robertson ldquoOn the theory of univalent functionsrdquo Annalsof Mathematics vol 37 no 2 pp 374ndash408 1936

[11] A V Boyd ldquoCoefficient estimates for starlike functions of order120572rdquo Proceedings of the AmericanMathematical Society vol 17 pp1016ndash1018 1966

[12] H Silverman ldquoUnivalent functions with negative coefficientsrdquoProceedings of the American Mathematical Society vol 51 pp109ndash116 1975

[13] H M Srivastava and S Owa ldquoCertain subclasses of starlikefunctions Irdquo Journal ofMathematical Analysis and Applicationsvol 161 no 2 pp 405ndash415 1991

[14] K A Roper and T J Suffridge ldquoConvex mappings on the unitball ofC119899rdquo Journal drsquoAnalyseMathematique vol 65 pp 333ndash3471995

[15] I Graham and G Kohr ldquoUnivalent mappings associated withthe Roper-Suffridge extension operatorrdquo Journal drsquoAnalyseMathematique vol 81 pp 331ndash342 2000

[16] S Gong and T S Liu ldquoThe generalized Roper-Suffridge exten-sion operatorrdquo Journal of Mathematical Analysis and Applica-tions vol 284 no 2 pp 425ndash434 2003

[17] G Kohr ldquoLoewner chains and a modification of the Roper-Suffridge extension operatorrdquo Mathematica vol 71 no 1 pp41ndash48 2006

[18] J R Muir ldquoA class of Loewner chain preserving extensionoperatorsrdquo Journal of Mathematical Analysis and Applicationsvol 337 no 2 pp 862ndash879 2008

[19] Y H Zhao Almost Starlike Mappings of Complex Order 120582 onthe Unit Ball of a Complex Banach Space Zhejiang NormalUniversity Jinhua China 2013

[20] L V Ahlfors Complex Analysis McGraw-Hill New York NYUSA 3rd edition 1978

[21] L Bieberbach ldquoUber einige extremal problem in gebiete derkonformen abbildungrdquo Mathematische Annalen vol 77 no 2pp 153ndash172 1916

[22] T Hayami S Owa and H M Srivastava ldquoCoefficient inequal-ities for certain classes of analytic and univalent functionsrdquoJournal of Inequalities in Pure and Applied Mathematics vol 8no 4 pp 1ndash10 2007

[23] M S Liu and Y C Zhu ldquoGeneralized Roper-Suffridge oper-ators on bounded and complete Reinhardt domainsrdquo Sciencein China vol 37 no 10 pp 1193ndash1206 2007

[24] Y Y Cui and C J Wang ldquoThe generalized Roper-Suffridgeextension operators in complex Banach spacerdquo Journal ofMathematics vol 34 no 3 pp 515ndash520 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of