research article coefficient estimates for certain classes of bi … · 2019. 7. 31. ·...

Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2013 Article ID 190560 4 pageshttpdxdoiorg1011552013190560

Research ArticleCoefficient Estimates for Certain Classes ofBi-Univalent Functions

Jay M Jahangiri1 and Samaneh G Hamidi2

1 Department of Mathematical Sciences Kent State University Burton OH 44021-9500 USA2 Institute of Mathematical Sciences Faculty of Science University of Malaya 50603 Kuala Lumpur Malaysia

Correspondence should be addressed to Jay M Jahangiri jjahangikentedu

Received 29 April 2013 Revised 27 July 2013 Accepted 31 July 2013

Academic Editor Heinrich Begehr

Copyright copy 2013 J M Jahangiri and S G Hamidi This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

A function analytic in the open unit disk D is said to be bi-univalent in D if both the function and its inverse map are univalentthere The bi-univalency condition imposed on the functions analytic in Dmakes the behavior of their coefficients unpredictableNotmuch is known about the behavior of the higher order coefficients of classes of bi-univalent functionsWe use Faber polynomialexpansions of bi-univalent functions to obtain estimates for their general coefficients subject to certain gap series aswell as providingbounds for early coefficients of such functions

1 Introduction

Let A denote the class of functions 119891 which are analytic inthe open unit disk D = 119911 isin C |119911| lt 1 and normalized by

119891 (119911) = 119911 +infin

sum119899=2

119886119899119911119899 (1)

Let S denote the class of functions 119891 isin A that areunivalent in D and let P be the class of functions 119901(119911) =1 + sum

infin

119899=1119901119899119911119899 that are analytic in D and satisfy the condition

Re(119901(119911)) gt 0 in D By the Caratheodory lemma (eg see [1])we have |119901

119899| le 2

For 0 le 120572 lt 1 and 120582 ge 1 we letD(120572 120582) denote the familyof analytic functions 119891 isin A so that

Re((1 minus 120582)119891 (119911)

119911+ 1205821198911015840 (119911)) gt 120572 119911 isin D (2)

We note that D(0 1) is the class of bounded boundaryturning functions and also thatD(120572 120582) sub D(120573 120582) if 0 le 120573 lt120572 For 119891 isin A the classD(120572 120582) sub S and was first defined andinvestigated by Ding et al [2]

It is well known that every function 119891 isin S has an inverse119891minus1 satisfying 119891minus1(119891(119911)) = 119911 for 119911 isin D and 119891(119891minus1(119908)) = 119908

for |119908| lt 14 according to Kobe One QuarterTheorem (egsee [1])

A function 119891 isin A is said to be bi-univalent in D ifboth 119891 isin S and 119892 = 119891minus1 isin S Finding bounds for thecoefficients of classes of bi-univalent functions dates backto 1967 (see Lewin [3]) But the interest on the bounds forthe coefficients of classes of bi-univalent functions pickedup by the publications of Brannan and Taha [4] Srivastavaet al [5] Frasin and Aouf [6] Ali et al [7] and Hamidi etal [8]The bi-univalency condition imposed on the functions119891 isin Amakes the behavior of their coefficients unpredictableNot much is known about the behavior of the higher ordercoefficients of classes of bi-univalent functions as Ali et al[7] also remarked that finding the bounds for |119886

119899| when 119899 ge 4

is an open problem Here in this paper we let119891 isin D(120572 120582) and119892 = 119891minus1 isin D(120572 120582) and use the Faber polynomial coefficientexpansions to provide bounds for the general coefficients |119886

119899|

of such functions with a given gap series We also obtainestimates for the first two coefficients |119886

2| and |119886

3| of these

functions as well as providing an estimate for their coefficientbody (119886

2 1198863) The bounds provided in this paper prove to be

better than those estimates provided by Srivastava et al [5]and Frasin and Aouf [6]

2 International Journal of Mathematics and Mathematical Sciences

2 Main Results

Using the Faber polynomial expansion of functions 119891 isin A ofthe form (1) the coefficients of its inverse map 119892 = 119891minus1 maybe expressed as [9]

119892 (119908) = 119891minus1 (119908) = 119908 +infin

sum119899=2

1

119899119870minus119899119899minus1

(1198862 1198863 ) 119908119899 (3)

where

119870minus119899119899minus1

=(minus119899)

(minus2119899 + 1) (119899 minus 1)119886119899minus12

+(minus119899)

(2 (minus119899 + 1)) (119899 minus 3)119886119899minus32

1198863

+(minus119899)


1198864

+(minus119899)


[1198865

+ (minus119899 + 2) 11988623]

+(minus119899)


[1198866

+ (minus2119899 + 5) 11988631198864]

+ sum119895ge7

119886119899minus119895

2119881119895

(4)

such that 119881119895with 7 le 119895 le 119899 is a homogeneous polynomial

in the variables 1198862 1198863 119886

119899[10] In particular the first three

terms of 119870minus119899119899minus1

are

1

2119870minus21

= minus1198862

1

3119870minus32

= 211988622

minus 1198863

1

4119870minus43

= minus (511988632

minus 511988621198863

+ 1198864)

(5)

In general for any 119901 isin N an expansion of 119870119901119899is as [9 page

183]

119870119901119899

= 119901119886119899

+119901 (119901 minus 1)

21198632119899

+119901

(119901 minus 3)31198633119899

+ sdot sdot sdot +119901

(119901 minus 119899)119899119863119899119899

(6)

where 119863119901119899

= 119863119901119899

(1198862 1198863 ) and by [11] or [12]

119863119898119899

(1198861 1198862 119886

119899) =infin

sum119898=1

119898(1198861)1205831 sdot sdot sdot (119886

119899)120583119899

1205831 sdot sdot sdot 120583119899

(7)

while 1198861

= 1 and the sum is taken over all nonnegativeintegers 120583

1 120583

119899satisfying

1205831

+ 1205832

+ sdot sdot sdot + 120583119899

= 119898

1205831

+ 21205832

+ sdot sdot sdot + 119899120583119899

= 119899(8)

Evidently 119863119899119899(1198861 1198862 119886

119904+119898) = 1198861198991 [13]

Theorem 1 For 0 le 120572 lt 1 and 120582 ge 1 let 119891 isin D(120572 120582) and119892 isin D(120572 120582) If 119886

119896= 0 2 le 119896 le 119899 minus 1 then

10038161003816100381610038161198861198991003816100381610038161003816 le

2 (1 minus 120572)

1 + (119899 minus 1) 120582 119899 ge 4 (9)

Proof For analytic functions 119891 of the form (1) we have

(1 minus 120582)119891 (119911)

119911+ 1205821198911015840 (119911)

= 1 +infin

sum119899=2

(1 + (119899 minus 1) 120582) 119886119899119911119899minus1

(10)

and for its inverse map 119892 = 119891minus1 we have

(1 minus 120582)119892 (119908)

119908+ 1205821198921015840 (119908)

= 1 +infin

sum119899=2

(1 + (119899 minus 1) 120582) 119887119899119908119899minus1

= 1 +infin

sum119899=2

(1 + (119899 minus 1) 120582)

times1

119899119870minus119899119899minus1

(1198862 1198863 119886

119899) 119908119899minus1

(11)

On the other hand since 119891 isin D(120572 120582) and 119892 = 119891minus1 isinD(120572 120582) by definition there exist two positive real partfunctions 119901(119911) = 1 + sum

infin

119899=1119888119899119911minus119899 and 119902(119908) = 1 + sum

infin

119899=1119889119899119908minus119899

where Re119901(119911) gt 0 and Re 119902(119908) gt 0 inD so that

(1 minus 120582)119891 (119911)

119911+ 1205821198911015840 (119911)

= 1 + (1 minus 120572)infin

sum119899=1

1198701119899

(1198881 1198882 119888

119899) 119911119899

(12)

(1 minus 120582)119892 (119908)

119908+ 1205821198921015840 (119908)

= 1 + (1 minus 120572)infin

sum119899=1

1198701119899

(1198891 1198892 119889

119899) 119908119899

(13)

Comparing the corresponding coefficients of (10) and (12)yields

(1 + 120582 (119899 minus 1)) 119886119899

= (1 minus 120572) 1198701119899minus1

(1198881 1198882 119888

119899minus1) (14)

and similarly from (11) and (13) we obtain

1

119899(1 + (119899 minus 1) 120582) 119870minus119899

119899minus1(1198870 1198871 119887

119899)

= (1 minus 120572) 1198701119899minus1

(1198891 1198892 119889

119899minus1)

(15)

Note that for 119886119896

= 0 2 le 119896 le 119899 minus 1 we have 119887119899

= minus119886119899and so

(1 + (119899 minus 1) 120582) 119886119899

= (1 minus 120572) 119888119899minus1

minus (1 + (119899 minus 1) 120582) 119886119899

= (1 minus 120572) 119889119899minus1

(16)

International Journal of Mathematics and Mathematical Sciences 3

Now taking the absolute values of either of the above twoequations and applying the Caratheodory lemma we obtain

10038161003816100381610038161198861198991003816100381610038161003816 le

(1 minus 120572)1003816100381610038161003816119888119899minus1

1003816100381610038161003816|1 + (119899 minus 1) 120582|

=(1 minus 120572)

1003816100381610038161003816119889119899minus11003816100381610038161003816

|1 + (119899 minus 1) 120582|le

2 (1 minus 120572)

1 + (119899 minus 1) 120582

(17)

Theorem 2 For 0 le 120572 lt 1 and 120582 ge 1 let 119891 isin D(120572 120582) and119892 isin D(120572 120582) Then one has the following

(i) 100381610038161003816100381611988621003816100381610038161003816 le

radic2 (1 minus 120572)

1 + 2120582 0 le 120572 lt

1 + 2120582 minus 1205822

2 (1 + 2120582)

2 (1 minus 120572)

1 + 120582

1 + 2120582 minus 1205822

2 (1 + 2120582)le 120572 lt 1

(ii) 100381610038161003816100381611988631003816100381610038161003816 le

2 (1 minus 120572)

1 + 2120582

(iii) 100381610038161003816100381610038161198863 minus 211988622

10038161003816100381610038161003816 le2 (1 minus 120572)

1 + 2120582

(18)

Proof Replacing 119899 by 2 and 3 in (14) and (15) respectively wededuce

(1 + 120582) 1198862

= (1 minus 120572) 1198881 (19)

(1 + 2120582) 1198863

= (1 minus 120572) 1198882 (20)

minus (1 + 120582) 1198862

= (1 minus 120572) 1198891 (21)

(1 + 2120582) (211988622

minus 1198863) = (1 minus 120572) 119889

2 (22)

Dividing (19) or (21) by (1 + 120582) taking their absolutevalues and applying the Caratheodory lemma we obtain

100381610038161003816100381611988621003816100381610038161003816 le

(1 minus 120572)10038161003816100381610038161198881

10038161003816100381610038161 + 120582

=(1 minus 120572)

100381610038161003816100381611988911003816100381610038161003816

1 + 120582le

2 (1 minus 120572)

1 + 120582 (23)

Adding (20) to (22) implies

2 (1 + 2120582) 11988622

= (1 minus 120572) (1198882

+ 1198892) (24)

or

11988622

=(1 minus 120572) (119888

2+ 1198892)

2 (1 + 2120582) (25)

An application of Caratheodory lemma followed bytaking the square roots yields

100381610038161003816100381611988621003816100381610038161003816 le

2radic(1 minus 120572) (

100381610038161003816100381611988821003816100381610038161003816 +

100381610038161003816100381611988921003816100381610038161003816)

2 (1 + 2120582)le2radic

2 (1 minus 120572)

1 + 2120582 (26)

Now the bounds given in Theorem 2 (i) for |1198862| follow

upon noting that if (1 + 2120582 minus 1205822)2(1 + 2120582) le 120572 lt 1 then

2 (1 minus 120572)

1 + 120582le2radic

2 (1 minus 120572)

1 + 2120582 (27)

Dividing (20) by (1 + 2120582) taking the absolute values of bothsides and applying the Caratheodory lemma yield

100381610038161003816100381611988631003816100381610038161003816 =

(1 minus 120572)10038161003816100381610038161198882

10038161003816100381610038161 + 2120582

le2 (1 minus 120572)

1 + 2120582 (28)

Dividing (22) by (1 + 2120582) taking the absolute values of bothsides and applying the Caratheodory lemma we obtain

100381610038161003816100381610038161198863 minus 211988622

10038161003816100381610038161003816 le2 (1 minus 120572)

1 + 2120582 (29)

Corollary 3 For 0 le 120572 lt 1 let 119891 isin D(120572 1) and 119892 isin D(120572 1)Then one has the following

(i) 100381610038161003816100381611988621003816100381610038161003816 le


3 0 le 120572 lt

1

3

1 minus 1205721

3le 120572 lt 1

(ii) 100381610038161003816100381611988631003816100381610038161003816 le

2 (1 minus 120572)

3

(30)

Remark 4 The above two estimates for |1198862| and |119886

3| show that

the bounds given in Theorem 2 are better than those givenby Srivastava et al ([5 page 1191 Theorem 2] and Frasin andAouf [6 page 1572 Theorem 32])

References

[1] P L Duren Univalent Functions vol 259 of Grundlehren derMathematischenWissenschaften Springer New York NY USA1983

[2] S S Ding Y Ling and G J Bao ldquoSome properties of a classof analytic functionsrdquo Journal of Mathematical Analysis andApplications vol 195 no 1 pp 71ndash81 1995

[3] M Lewin ldquoOn a coefficient problem for bi-univalent functionsrdquoProceedings of the American Mathematical Society vol 18 pp63ndash68 1967

[4] D A Brannan and T S Taha ldquoOn some classes of bi-univalentfunctionsrdquo Studia Universitatis Babes-Bolyai Mathematica vol31 no 2 pp 70ndash77 1986

[5] H M Srivastava A K Mishra and P Gochhayat ldquoCertainsubclasses of analytic and bi-univalent functionsrdquo AppliedMathematics Letters vol 23 no 10 pp 1188ndash1192 2010

[6] B A Frasin and M K Aouf ldquoNew subclasses of bi-univalentfunctionsrdquoAppliedMathematics Letters vol 24 no 9 pp 1569ndash1573 2011

[7] R M Ali S K Lee V Ravichandran and S SupramaniamldquoCoefficient estimates for bi-univalent Ma-Minda starlike andconvex functionsrdquo Applied Mathematics Letters vol 25 no 3pp 344ndash351 2012

[8] S G Hamidi S A Halim and J M Jahangiri ldquoFaber poly-nomial coefficient estimates for meromorphic bi-starlike func-tionsrdquo International Journal of Mathematics and MathematicalSciences vol 2013 Article ID 498159 4 pages 2013

[9] H Airault and A Bouali ldquoDifferential calculus on the Faberpolynomialsrdquo Bulletin des Sciences Mathematiques vol 130 no3 pp 179ndash222 2006


[10] H Airault and J Ren ldquoAn algebra of differential operators andgenerating functions on the set of univalent functionsrdquo Bulletindes Sciences Mathematiques vol 126 no 5 pp 343ndash367 2002

[11] P G Todorov ldquoOn the Faber polynomials of the univalentfunctions of class Σrdquo Journal of Mathematical Analysis andApplications vol 162 no 1 pp 268ndash276 1991

[12] H Airault ldquoSymmetric sums associated to the factorization ofGrunsky coefficientsrdquo in Conference Groups and SymmetriesMontreal Canada April 2007

[13] H Airault ldquoRemarks on Faber polynomialsrdquo InternationalMathematical Forum Journal for Theory and Applications vol3 no 9-12 pp 449ndash456 2008

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Stochastic AnalysisInternational Journal of


2 Main Results

Using the Faber polynomial expansion of functions 119891 isin A ofthe form (1) the coefficients of its inverse map 119892 = 119891minus1 maybe expressed as [9]

119892 (119908) = 119891minus1 (119908) = 119908 +infin

sum119899=2

1

119899119870minus119899119899minus1

(1198862 1198863 ) 119908119899 (3)

where

119870minus119899119899minus1

=(minus119899)


+(minus119899)


1198863

+(minus119899)


1198864

+(minus119899)


[1198865

+ (minus119899 + 2) 11988623]

+(minus119899)


[1198866

+ (minus2119899 + 5) 11988631198864]

+ sum119895ge7

119886119899minus119895

2119881119895

(4)

such that 119881119895with 7 le 119895 le 119899 is a homogeneous polynomial

in the variables 1198862 1198863 119886

119899[10] In particular the first three

terms of 119870minus119899119899minus1

are

1

2119870minus21

= minus1198862

1

3119870minus32

= 211988622

minus 1198863

1

4119870minus43

= minus (511988632

minus 511988621198863

+ 1198864)

(5)

In general for any 119901 isin N an expansion of 119870119901119899is as [9 page

183]

119870119901119899

= 119901119886119899

+119901 (119901 minus 1)

21198632119899

+119901

(119901 minus 3)31198633119899

+ sdot sdot sdot +119901

(119901 minus 119899)119899119863119899119899

(6)

where 119863119901119899

= 119863119901119899

(1198862 1198863 ) and by [11] or [12]

119863119898119899

(1198861 1198862 119886

119899) =infin

sum119898=1

119898(1198861)1205831 sdot sdot sdot (119886

119899)120583119899

1205831 sdot sdot sdot 120583119899

(7)

while 1198861

= 1 and the sum is taken over all nonnegativeintegers 120583

1 120583

119899satisfying

1205831

+ 1205832

+ sdot sdot sdot + 120583119899

= 119898

1205831

+ 21205832

+ sdot sdot sdot + 119899120583119899

= 119899(8)

Evidently 119863119899119899(1198861 1198862 119886

119904+119898) = 1198861198991 [13]

Theorem 1 For 0 le 120572 lt 1 and 120582 ge 1 let 119891 isin D(120572 120582) and119892 isin D(120572 120582) If 119886

119896= 0 2 le 119896 le 119899 minus 1 then

10038161003816100381610038161198861198991003816100381610038161003816 le

2 (1 minus 120572)

1 + (119899 minus 1) 120582 119899 ge 4 (9)

Proof For analytic functions 119891 of the form (1) we have

(1 minus 120582)119891 (119911)

119911+ 1205821198911015840 (119911)

= 1 +infin

sum119899=2

(1 + (119899 minus 1) 120582) 119886119899119911119899minus1

(10)

and for its inverse map 119892 = 119891minus1 we have

(1 minus 120582)119892 (119908)

119908+ 1205821198921015840 (119908)

= 1 +infin

sum119899=2

(1 + (119899 minus 1) 120582) 119887119899119908119899minus1

= 1 +infin

sum119899=2

(1 + (119899 minus 1) 120582)

times1

119899119870minus119899119899minus1

(1198862 1198863 119886

119899) 119908119899minus1

(11)

On the other hand since 119891 isin D(120572 120582) and 119892 = 119891minus1 isinD(120572 120582) by definition there exist two positive real partfunctions 119901(119911) = 1 + sum

infin

119899=1119888119899119911minus119899 and 119902(119908) = 1 + sum

infin

119899=1119889119899119908minus119899

where Re119901(119911) gt 0 and Re 119902(119908) gt 0 inD so that

(1 minus 120582)119891 (119911)

119911+ 1205821198911015840 (119911)

= 1 + (1 minus 120572)infin

sum119899=1

1198701119899

(1198881 1198882 119888

119899) 119911119899

(12)

(1 minus 120582)119892 (119908)

119908+ 1205821198921015840 (119908)

= 1 + (1 minus 120572)infin

sum119899=1

1198701119899

(1198891 1198892 119889

119899) 119908119899

(13)

Comparing the corresponding coefficients of (10) and (12)yields

(1 + 120582 (119899 minus 1)) 119886119899

= (1 minus 120572) 1198701119899minus1

(1198881 1198882 119888

119899minus1) (14)

and similarly from (11) and (13) we obtain

1

119899(1 + (119899 minus 1) 120582) 119870minus119899

119899minus1(1198870 1198871 119887

119899)

= (1 minus 120572) 1198701119899minus1

(1198891 1198892 119889

119899minus1)

(15)

Note that for 119886119896

= 0 2 le 119896 le 119899 minus 1 we have 119887119899

= minus119886119899and so

(1 + (119899 minus 1) 120582) 119886119899

= (1 minus 120572) 119888119899minus1

minus (1 + (119899 minus 1) 120582) 119886119899

= (1 minus 120572) 119889119899minus1

(16)



10038161003816100381610038161198861198991003816100381610038161003816 le

(1 minus 120572)1003816100381610038161003816119888119899minus1

1003816100381610038161003816|1 + (119899 minus 1) 120582|

=(1 minus 120572)

1003816100381610038161003816119889119899minus11003816100381610038161003816

|1 + (119899 minus 1) 120582|le

2 (1 minus 120572)

1 + (119899 minus 1) 120582

(17)


(i) 100381610038161003816100381611988621003816100381610038161003816 le


1 + 2120582 0 le 120572 lt

1 + 2120582 minus 1205822

2 (1 + 2120582)

2 (1 minus 120572)

1 + 120582

1 + 2120582 minus 1205822

2 (1 + 2120582)le 120572 lt 1

(ii) 100381610038161003816100381611988631003816100381610038161003816 le

2 (1 minus 120572)

1 + 2120582

(iii) 100381610038161003816100381610038161198863 minus 211988622

10038161003816100381610038161003816 le2 (1 minus 120572)

1 + 2120582

(18)


(1 + 120582) 1198862

= (1 minus 120572) 1198881 (19)

(1 + 2120582) 1198863

= (1 minus 120572) 1198882 (20)

minus (1 + 120582) 1198862

= (1 minus 120572) 1198891 (21)

(1 + 2120582) (211988622

minus 1198863) = (1 minus 120572) 119889

2 (22)


100381610038161003816100381611988621003816100381610038161003816 le

(1 minus 120572)10038161003816100381610038161198881

10038161003816100381610038161 + 120582

=(1 minus 120572)

100381610038161003816100381611988911003816100381610038161003816

1 + 120582le

2 (1 minus 120572)

1 + 120582 (23)


2 (1 + 2120582) 11988622

= (1 minus 120572) (1198882

+ 1198892) (24)

or

11988622

=(1 minus 120572) (119888

2+ 1198892)

2 (1 + 2120582) (25)


100381610038161003816100381611988621003816100381610038161003816 le


100381610038161003816100381611988821003816100381610038161003816 +

100381610038161003816100381611988921003816100381610038161003816)

2 (1 + 2120582)le2radic

2 (1 minus 120572)

1 + 2120582 (26)



2 (1 minus 120572)

1 + 120582le2radic

2 (1 minus 120572)

1 + 2120582 (27)


100381610038161003816100381611988631003816100381610038161003816 =

(1 minus 120572)10038161003816100381610038161198882

10038161003816100381610038161 + 2120582

le2 (1 minus 120572)

1 + 2120582 (28)


100381610038161003816100381610038161198863 minus 211988622

10038161003816100381610038161003816 le2 (1 minus 120572)

1 + 2120582 (29)


(i) 100381610038161003816100381611988621003816100381610038161003816 le


3 0 le 120572 lt

1

3

1 minus 1205721

3le 120572 lt 1

(ii) 100381610038161003816100381611988631003816100381610038161003816 le

2 (1 minus 120572)

3

(30)


3| show that


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











10038161003816100381610038161198861198991003816100381610038161003816 le

(1 minus 120572)1003816100381610038161003816119888119899minus1

1003816100381610038161003816|1 + (119899 minus 1) 120582|

=(1 minus 120572)

1003816100381610038161003816119889119899minus11003816100381610038161003816

|1 + (119899 minus 1) 120582|le

2 (1 minus 120572)

1 + (119899 minus 1) 120582

(17)


(i) 100381610038161003816100381611988621003816100381610038161003816 le


1 + 2120582 0 le 120572 lt

1 + 2120582 minus 1205822

2 (1 + 2120582)

2 (1 minus 120572)

1 + 120582

1 + 2120582 minus 1205822

2 (1 + 2120582)le 120572 lt 1

(ii) 100381610038161003816100381611988631003816100381610038161003816 le

2 (1 minus 120572)

1 + 2120582

(iii) 100381610038161003816100381610038161198863 minus 211988622

10038161003816100381610038161003816 le2 (1 minus 120572)

1 + 2120582

(18)


(1 + 120582) 1198862

= (1 minus 120572) 1198881 (19)

(1 + 2120582) 1198863

= (1 minus 120572) 1198882 (20)

minus (1 + 120582) 1198862

= (1 minus 120572) 1198891 (21)

(1 + 2120582) (211988622

minus 1198863) = (1 minus 120572) 119889

2 (22)


100381610038161003816100381611988621003816100381610038161003816 le

(1 minus 120572)10038161003816100381610038161198881

10038161003816100381610038161 + 120582

=(1 minus 120572)

100381610038161003816100381611988911003816100381610038161003816

1 + 120582le

2 (1 minus 120572)

1 + 120582 (23)


2 (1 + 2120582) 11988622

= (1 minus 120572) (1198882

+ 1198892) (24)

or

11988622

=(1 minus 120572) (119888

2+ 1198892)

2 (1 + 2120582) (25)


100381610038161003816100381611988621003816100381610038161003816 le


100381610038161003816100381611988821003816100381610038161003816 +

100381610038161003816100381611988921003816100381610038161003816)

2 (1 + 2120582)le2radic

2 (1 minus 120572)

1 + 2120582 (26)



2 (1 minus 120572)

1 + 120582le2radic

2 (1 minus 120572)

1 + 2120582 (27)


100381610038161003816100381611988631003816100381610038161003816 =

(1 minus 120572)10038161003816100381610038161198882

10038161003816100381610038161 + 2120582

le2 (1 minus 120572)

1 + 2120582 (28)


100381610038161003816100381610038161198863 minus 211988622

10038161003816100381610038161003816 le2 (1 minus 120572)

1 + 2120582 (29)


(i) 100381610038161003816100381611988621003816100381610038161003816 le


3 0 le 120572 lt

1

3

1 minus 1205721

3le 120572 lt 1

(ii) 100381610038161003816100381611988631003816100381610038161003816 le

2 (1 minus 120572)

3

(30)


3| show that


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article coefficient estimates for certain classes of bi … · 2019. 7. 31. ·...

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