Proceedings ofthe SEAMS-GMU Conference 2003

The Coefficients of Functions with Positive Real Partand Some Special Classes of Univalent Functions

Rosihan M. AliSchool of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia

e-mail: rosihan({iks.usrn.my

PREFACE

The SEAMS-Gadjah Mada University International Conference 2003 onMathematics and Its Applications was held on 14 - 17 July 2003 at Gadjah MadaUniversity, Yogyakarta, Indonesia. The Conference is the forth conference held byGadjah Mada University and SEAMS. The former was held in 1989, 1995 and 1999.

The Conference has achieved its main purposes of promoting the exchange ofideas and presentation of recent development, particularly in the areas of pure andapplied mathematics, which are represented in South East Asian Countries. TheConference has also provided a forum of researchers, developers, and practitioners toexchange ideas and to discuss future direction of research. Moreover, it has enhancedcollaboration between researchers from countries in the region and those fromoutside.

During the 4-day conference there were 13 plenary lectures and 117contributed papers communications. The plenary lectures were delivered by Prof.Chew Tuan Seng (Singapore), Prof. Edy Soewono (Indonesia), Prof. D. K. Ganguly(India), Prof. F. Kappel (Austria), Prof G. Desch (Austria), Prof. G. Peichl (Austria),Prof. J. A. M. van der Weide (the Netherland), Prof. K. Denecke (Germany), Prof.Lee Peng Vee (Singapore), Prof. Soeparna Darmawijaya (Indonesia), Prof. SuthepSuantai (Thailand), Prof. V. Dlab (Canada) and Dr. Widodo (Indonesia). Most of thecontributed papers were delivered by Mathematicians from Asia.

The proceedings consists of 5 invited lectures and 64 refereed contributedpapers.

In this occasion, we would like to express our gratitute and appreciation to thefollowing sponsors:

• UNESCO Jakarta• ASEA UNINET• ICTP• BANK MANDIRI• Gadjah Mada University• Faculty of Mathematics and Natural Sciences, Gadjah Mada University• Department of Mathematics, Gadjah Mada University

for their assistance and support.

iii

We would like to extend our appreCIatIOn to the invited speakers, theparticipants and the referees for the wonderful cooperation, the great coordination andthe fascinating effort. We would like to thank to our colleagues who help in editingpapers especially to Atok Zuliyanto, Imam Sholekhudin, Fajar Adikusumo, I GedeMujiyatna and Sri Haryatmi. Finally, we would like to acknowledge and express ourthanks for the help and support of the staff and friends in the MathematicsDepartment, UGM in the preparation for and during the conference.

Editorial BoardLina AryatiSupamaBudi SurodjoCh. Rini Indrati

iv

CONTENTS

Cover PagePrefaceContents

INVITED SPEAKER PAPERSOptimization for Large Gas Transmission Network Using Least DifferenceHeuristic AlgorithmEdy Soewono, Septoratno Siregar, Inna Savitri Widiasri and Nadia Ariani

An Immersed Interface Technique for Mixed Boundary Value ProblemsGunther Peichl

Interaction between Mathematics and FinanceHans van der Weide

On the Bounded Variation Interval FunctionsSoeparna Darmawijaya

Topological Entropy of Discrete Dynamical SystemWidodo

SELECTED CONTRIBUTED PAPERS

1ll

V

14

24

33

51

Part I: AlgebraConvergence of the Divide-et-impera Algorithm in Case ofa Decoupled 63SystemAri Suparwanto

Finiteness Conditions of Generalized Power Series Rings 75Budi Surodjo

The Application of Block Triangular Decoupling for Linear System over 83Principal Ideal Domains on the Pole AssignmentCaturiyati

Characterization of Fuzzy Group Actions in Terms of Group Homomorphisms 93Frans Susilo

On Cyclic N-ary Gray Codes 98I Nengah Suparta and A. 1. van Zanten

v

The Properties of Non-Degenerate Bilinear Forms 106Karyati and Sri Wahyuni

An Extremal Problem on the Number ofYertices of Minimum Degree in 112Critically Connected GraphsKetut Budayasa

A Non Reductionist, Non Deontical Philosophy of Mathematics 120Soehakso RMJT

On Maximal Sets of I-factors Having Disconnected Leaves of Even Deficiency 126Sugeng Mardiyono

Tabu Search Based Heuristics for the Degree Constrained Minimum Spanning 133Tree ProblemWamiliana and Louis Caccetta

Necessary Conditions for The Existence of (d, k)-Digraphs Containing 141SelfrepeatsY.M. Cholily and E. T. Baskoro

Part II: Analysis

Essentially Small Riemann Sum in the n-Dimensional Space 148Cit. Rini Indrati

Bilipschitz Triviality Does Not Imply Lipschtiz Regularity 155Dwi Juniati

Analytic Geometry in the Real Hilbert Space 160Imam Solekhudin and Soeparna Darmawijaya

The Henstock Integral in a Locally Compact Metric Space for Vector Valued 166FunctionsManuharawati

The Coefficients of Functions with Positive Real Part and Some Special Classes 173of Univalent FunctionsRosihan M. Ali

(J' - Order Continuous and Ortogonally Additive Operators On a Space of 185Banach Lattice Valued FunctionsSupama and Soeparna Darmawijaya

VI

Reserve-Cesaro-Orlicz Sequence Spaces and Their Second DualsY. M. Sri Daru Unoningsih

On the Henstock Integral ofFunctions Defined on R n with Values in lP­

spaces, 1~ p < ex)

Zahriwan, Soeparna Darmawijaya and Y. M. Sri Daru Unoningsih

Part III: Applied Mathematics

192

200

Critical Vaccination Level for Dengeu Fever Disease Transmission 208Asep K. Supriatna and Edy Soewono

The Dynamics of Numerical Methods for Stochastic Differential Equations 218Revina D. Handari

Modeling of Wood Drying 227Edi Cahyono. Tjang Daniel Chandra and La Gubu

The application of Implisit Kalman Filtering on A One Dimensional Shallow 234Water ProblemErna Apriliani and A. W. Heemink

Bifurcation of Parametrically Excited van der Pol Equation 241Fajar Adi Kusumo. Wono Setya Budhi and J. M. Tuwankotta

Waveform Relaxation Techniques for Stochastic Differential Equations with 248Additive NoiseGatot F. Hertono

Higher Order Averaging in Linear Systems and an Application 256Hartono

Internal Solitary Waves in the Lombok Strait 265Jaharudin, S. R. Pudjaprasetya and Andonowati

Determination of Initial Condition for Asymptotic Stability of Extended 277Systems (Nonlinear Control Systems and Direct Gradient Descent Control)Janson Naiborhu

Application of Optimal Control to Forecast Indonesian Economic Growth 285Johanna Endah Louise, Ponidi and Martin Panggabean

A Semi-discrete Wavelet Collocation Method for Parabolic Equation 293Julan Hernadi. Gunther Peichl and Bambang Soedijono

vii

Perturbation of a Discrete Generalized Eigenvalue 303Lina Aryati

A Level Set Method for Two-Phase Steam Displacing Water in A Saturated 309Porous MediumM. Muksar, E. Soewono and S. Siregar

The Effect of Vaccinations in a Dengue Fever Disease Transmission with Two- 321strain VirusesNuning Nuraini, Edy Soewono and Kuntjoro Adji Sidarto

Exsistence of Nash Solution for Non Zero-Sum Linear Quadratic Game With 331Descriptor SystemSa/mah, S. M Nababan, Bambang S. and S. Wahyuni

A Branch and Cut for Single Commodity One Warehouse Inventory Routing 340ProblemSarwadi

A Finite Element Approach to Direct Current Resistivity Measurements in 349Aeolotropic Media- The Forward ProblemSri Mardiyati, Peg Foo Siew and Yong Hong Wu

Characteristic of Two Wave Group Interactions of an Improved Kdv Equation 358Sutimin and E. Soewono

Some Criteria for Formal Complete Integrability of Nonlinear Evolution 366EquationsTeguh Bharata Adji

Model Reduction of Linear Parameter Varying Systems 376Widowati, R. Saragih, Riyanto Bambang and S. M Nababan

Feasible Direction Technique for Quadratic Programming Problems In 2 And 3 384DimensionYosza Dasril, Mustafa Mamat and Ismail bin Mohd.

Construction and Reconstruction of 3D Face Images Uses Merging and 391Splitting Eigenspace ModelsYudi Satria and Benyamin Kusumoputro

VIII

Part IV: Computers Sciences

A Comparison Study of the Perfonnance of the Fourier Transfonn Based 400Algorithm and the Artificial Neural Network Based Algorithm in DetectingFabric Texture DefectAgus Harjoko, Sri Hartati and Dedi Trisnawarman

Development of 3D Authoring Tool Based on VRML 407Bachtiar Anwar

Right Inverse in Public Key Cryptosystem Design 414Budi Murtiyasa, Subanar, Retantyo Wardoyo and Sri Hartati

A Study on Analyzing Texture Using Walsh Transfonn 419Jung-yun Hur and Kyoung-Doo Son

A study of High-Availability Disk Chace Manager for Distributed Shared- 428Disks Using a Genetic AlgorithmPall-am, Han and Kwang-Youn, Jin

Evaluating Land Suitability using Fuzzy Logic Method 437Sri Harta/i, Agus Harjoko and [mas Sukaesih Sitanggang

Iris Recognition System using Hadamard Transfonnation 446Lee Xuan Truong

Part V: Statistics

Interval Estimation for Survivor Function on Two Parameters Exponential 452Distribution under Multiple Type-II Cencoring on Simple Case with BootstrrapPercentileAkhmad Fauzy, Noor Akma Ibrahim, Isa Daud and Mohd. Rizam Abu Bakar

The Stationary Conditions OfThe Generalized Space-Time Autoregressive 460ModelBudi Nurani Ruclljana

Realibility Analysis of k-out-of-n Systems using Generalized Lambda 466DistributionsHa.\)'im Gautama and A. 1. C. van Gemund

Regularized Orthogonal Least Squares-based Radial Basis Function Networks 475and Its Application on Time Series PredictionHendri Murfi and Djati Kerami

IX

Analysis Of Relation Of Bernoulli Automorphism And Kolmogorov 484AutomorphismHenry Junus Wattimanela

Multivariate Outlier Detection using Mixture Model 489Juergen Franke and Bambang Susanto

Bandwith Choice for Kernel Density Estimation 494Kartiko

Statistical Properties of Indonesian Daily Stock Returns And Stochastic 500Volality ModelsKhreshna Syuhada and Gopalan Nair

Some Risk Measures for Capital Requirements in Actuarial Science 507Lienda Noviyanti and Muhammad Syamsuddin

The Intervals Estimation for Intercept using Bootstrap Methods (Case Of 515Bootstrap Persentile And BCA)Muhammad Safiih Lola and Norizan Mohamed

Bayesian Mixture Modeling in Reliability: MCMC Approaches 521Nur lriawan

Monte Carlo Approximation on Iterated Bootstrap Confidence Intervals 528Sri Haryatmi Kartiko

Partially Linear Model with Heteroscedastic Errors 533Sri Haryatmi Kartiko and Subanar

A Graph-Aided Method For Planning Two-Level Fractional Factorial 541Experiment When Certain Interactions Are Important Using Taguchi's LinearGraphsSuhardi Djojoatmodjo

Note on Renewal Reward Processes 551Suyono and J A. M. van der Weide

The Estimation Parameters In Space Time Autoregressive Moving Average X 557(STARMAX) Model For Missing DataUdjianna S. Pasaribu

x

Proceedings ofthe SEAMS-GMU Conference 2003

The Coefficients of Functions with Positive Real Partand Some Special Classes of Univalent Functions

Rosihan M. AliSchool of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia

e-mail: rosihan(£ilcs.usm.my

Abstract

Several special classes of univalent functions f in the unit disk U are characterized by

the quantity zf'(z)/I(z) lies in a given region in the right-half plane. Amongst these are

the classes SS· (a) of strongly starlike functions of order a and PS· (p) consisting of

parabolic starlike functions of order p. Both classes are closely related to the class P of

nonnalized analytic functions in U with positive real part.

We derive some sharp non-linear coefficient estimates for functions in the class P. Usingthese estimates, we detennine sharp bounds for the first four coefficients over the classesSS· (a) and PS· (p), and their inverses. All possible extremal functions are found. In

many of these problems, there cannot be a sole extremal function. The Fekete-Szegocoefficient functional is also treated.

Keywords: Univalent functions, analytic functions with positive real part, parabolic starlikefunctions. strongly starlike fu!,ctions. coefficient bound. Fekete-Szego coefficient functional.

1. Introduction

Let A denote the class of analytic functions f in the open unit disk U = {z:1 z 1< I}

and normalized so that f(O) =1'(0) - 1=O. Some special classes of univalent functions are

defined by natural geometric conditions. A well-known example is the class S· of starlikefunctions consisting of analytic functions f E A that map U conformally onto domains

starlike with respect to the origin O. Geometrically, this means that the linear segmentjoining 0 to every other point WE feU) lies entirely in feU).

Closely related to the class S· is the class P of normalized analytic functions p in

the unit disk U with positive real part such that p(O) =I and Re p(z) > 0, Z E U. It is known

[11,p.42Jthatafunction fEA belongs to s* ifandonlyif zf'(z)/f(Z)EP.

There are several subclasses of univalent starlike functions that are characterized bythe quantity zl'(z) / f(z) lies in a given region in the right-half plane. The region is often

convex and symmetric with respect to the real axis. Ma and Minda [8J have given a verygood unified treatment of such a study under a weaker condition that the region is starlikewith respect to I. We shall be interested in the following two subclasses.

173

Rosihan M. Ali

An analytic function f E A is said to be strongly starlike of order a, 0 < a:::; I, if it

satisfies

Izf'(z) 1fa

arg < - (z E U).f(z) 2

The set of all such functions is denoted by SS * (a). This class has been studied by several

authors [2,3,7,13,14]. More recently, Nunokawa and Owa [10] obtained a sufficient condition

for functions fEA to belong to SS*(a). With tp(z)=(:~;r, the class SS*(a) consists

offunctions f such that zf'(z)/ f(z) E tp(U), Z E U.For 0:::; p < 1, let n p be the parabolic region in the right-half plane

Q p = {w= u +iv: v2:::; 4(1- p)(u - p)}= {w:\ w-1\:::; 1- 2p + Rew}.

The class ofparabolic starlike functions oforder p is the subclass PS *(p) of A consisting

of functions f such that zf'(z)/ f(z) E n p , z E U. This class is a natural extension of the

class ofnorrnalized unifonnly convex functions UCV introduced by Goodman [4]. We recallthat a convex function f belongs to the class UCV if it has the additional property that for

every circular arc y contained in U with centre also in U, the image arc f(y) is convex. It

is known [9,12] that f EUCV ifandonly if zf'(Z)E Ps*(t)

If

is in the class SS * (a) (or PS *(p», then the inverse of f admits an expansion

f -1 ( 2 3w)=w+Y2w +Y3w +...

(I)

(2)

near w.= O. In this paper, we derive some sharp non-linear coefficient estimates for functionsin the class P. From these bounds, we detennine sharp bounds for the first four coefficients

of Ian lover both classes SS *(a) and PS *(p), the first four coefficients of Iyn lover

55 *(a), and find all possible extremal functions. Although the natural choice for an

extremal function would arise from p(z) = :~~ E P, we show that it cannot be the sole

extremal function for these problems. Additionally, we obtain sharp estimate for the Fekete­

Szego coefficient functionals la3 - ta2 21 or Iy3 - ty 221·

174

The Coefficients ofFunctions

2. Preliminary results

The classes SS *(a) and PS * (p) are closely related to the class P. It is clear that

f E SS· (a) if and only if there exists a function pEP so that zf'(z)/ f(z) = pa (z) . By

equating coefficients, each coefficient of f(z) = z + Q2Z2 + Q3Z3 + can be expressed in

tenns of coefficients of a function p(z) = I + clz + C2z2 + C3z3 + in the class P. For

example,

(3)

Using representations (1) and (2) together with f(f-I (w» = w or

w=f-'(w)+a2(f-I(w»2 +a3U-1(w»3 + ....

we obtain the relationships

(4)

Thus coefficient estimates for the class SS * (a) and its inverses may be considered as non­linear coefficient problems for the class P.

Turning to the class PS· (p), Ali and Singh [I] has shown that the nonnalized

Riemann mapping function q from U onto n p is given by

() I 4(1-P)[1 1+..[;]2q z = + og--1(2 1-..[;

If f(z) = z + b2z 2 + b3z 3 + ... E PS· (p), and h(z) = zf'(z)/ f(z), then there exists a

Schwarzian function w in U with w(O) = o. Iw(z) 1< I, and satisfying

zf'(z)h(z) =--=q(w(z».

I(z)(5)

175

Rosihan M. Ali

Hence the function

p(Z) = 1+q-I(h(z))

l-q-I(h(z))

is analytic and has positive real part in U, that is, pEP. It is now easily established that

Thus once again we see that coefficient estimates for PS *(p) may be viewed in- tenns of

non-linear coefficient problems for the class P. Our principal tool in these problems is givenin the following lemma.

Lemma 1 (5]. Afunction p(z) =I + Lk=1 Ck zk belongs to P ifand only if

~ {12z j + I CkZk+j!2 -I I ck+lzk+jI21~O)=0 k=1 k=O

for evelY sequence {Zk} ofcomplex numbers which sati!>!y limk-w;,suP Izk III k < 1.

I P 21 { 2,c2 --cl ::5:max{2,2Ip-lll=2 21,u -II,

0::5: p::5: 2

elsewhere

If p < 0 or p > 2, equality holds if and only if p(z) =(l +a)/(I- a), Ie 1= I. If 0 < p < 2,

then equality holds if and only if p(z) = (I + a 2 )/(1- a 2 ), Ie 1= I. For ,u = 0, equality

holds ifand only ifl+a I-a

p(z):=P2(z)=A.--+(I-A.)--, O::5:A.::5:I,lel=1.I-a l+a

For p =2, equality holds ifand only if p is the reciprocal of P2'

Remark. Ma and Minda [8] had earlier proved the above result. We give a different proof.

176

The Coefficients ofFunctions

Proof Choose the sequence {zd of complex numbers in Lemma 1 to be Zo = - f.JcIf2,

zi =I, and zk =0 if k > I. This yields

IC2 - ~CI2r + I ci 12 ~ 1(1- f.J)cd

2+4,

that is,

(7)

(8)

If f.J < 0 or f.J > 2, the expression on the right of inequality (7) is bounded above by

4(11-1)2. Equality holds ifand only if ICj 1=2, i.e., p(z)=(I+z)j(1-z; or its rotatiolls. If

0< f.J < 2, then the right expression of inequality (7) is bounded above by 4. In this case,

equality holds if and only if Icl 1= 0 and Ic2 1= 2, i.e., p(z) =(l + z2 )/0- z2) or its

rotations. Equality holds when f.J =0 if and only jf Ic2 1= 2, i.e., [I I, p. 41]

I+a I-ap(z):=P2(z)=,t--+(I-,t)--, O~A.:::;I, 1&1=1.

I-a I+a

Finally, when f.J = 2, then Ic2 - Cl2 I= 2 if and only if p is the reciprocal of P2' 0

Another interesting application of Lemma 1 occurs by choosing the sequence IZk}

to be 20 =be} - f3c2, Zl =-]1:'1, Z2 =I, and zk = 0 if k > 2. In this case, we find that

I'12 2 I 2 1

2, 21

2c3 -(fl+Y)Clc2 +ay' ~4+4y(y-l)lc! I +(28-y)c] -(2fl-l)c2 -(2 --J'C\ I

I 1

2 22 v 2 . (8 - fly) 4

=4+4y(y-I)lcll +4fJ(fJ-I)C2 --cl - ICII2 fJ(fJ -I)

where v'= 8(fJ -I) + fJ(8 - y). fJ(fJ-l) .

Lemma 3. Let p(z)=I+Lk=lckZk EP. !fO~fJ~1 and fJ(2f3-1)~8~fJ, then

IC3 - 2fJc(c2 + &13

/~ 2.

Proof If {J =0, then 8 =0 and the result follows since Ic3 I~ 2. If fJ = I, then Ii =1 and

the inequality follows from a result of[6].Now assume that 0 < fJ < 1 so that P(fJ - I) < O. With Y = {J, we find from (8) that

177

Rosihan M. Ali

~4+bx+CX2 :=h(x)

with X=lcI12e[0,4], b=4fJ(P-l), and c=-(t5-p 2)2/fJ (P-l). Since c:2:0, it follows

that h(x) ~ h(O) provided h(O) - h(4):2: 0, i.e., b + 4c ~ 0. This condition is equivalent to

18 - fJ2 I~ fJ(l- fJ), which completes the proof. 0

With t5 = P in Lemma 3, we obtain an extension of Libera and Zlotkiewicz result [6]

that IC3 - 2cI c2 + C13

\ ~ 2.

Corollary 1. If p(z)=I+Lk'=lckZk eP, and O~P~I, then

IC3 - 2f3c1 c2 + f3c1 31~ 2.

When P =0, equality holds ifand only if3 1+ &e-2trik /3z

p(Z):=P3(z) = L Ak -2trik/3' (/el=1)k=1 1- Ge Z

}'k :2: 0, with Al + A2 + A3 = I. If P = I, equality holds if and ollly if p is the reciprocal of

P3' If O<P<I, equality holds if and only if p(z)=(I+t:z)!I-cz),I&I=I, or

p(Z)=(I+cz 3)/(1-&z:3),l e l=1.

Proof We only need to find the extremal functions. If P = 0, then equality holds if and only

if IC3 1== 2, i.e., P is the function P3 [II, p. 41]. If P = I, then equality holds if and only if

p is the reciprocal of P3' When 0 < P< I, we deduce from (8) that

I 312 2 I 212 4c3 - 2f3cIC2 + f3c1 ~ 4 + 4fJ(fJ - I) lei 1 +4fJ(fJ - I) c2 --tCI - P(P - I) lei I

~4+4fJ(P-I)lcI12 _P(P_I)lcI1 4 ~ 4.

The bound 4 in the last inequality is obtained from simple calculus computations. Equality

occurs in the last inequality if and only if either Ici 1== 0 or 1ci 1= 2. If Ici 1= O. then 1c2 1= 0,

i.e., p(z)=(I+t:z 3 )/(I_cz3 ), 1&1=1. Iflcll=2, then p(z)=(I+t:z)!(1-t:z), 1&1=1. 0

Lemma 4. If p(z) = I + Ik=rCkZk E P, then

IC3 - (p + l)cl c2 + f.lC131~ max{2,212p -II}= { 2,2/2p-l/,

O~p~1

elsewhere

Proof For 0 ~ p ~ I, the estimate follows from Lemma 3 with t5 = p, and Zp = p + I. For the

second estimate, choose P= p, r = I, and t5 = p in (8). Since p(p-I) > 0, we conclude

178

The Coefficients ofFunctions

from (7) and (8) that

IC3 -(,u+l)clc2 +,uc1312~4+4,u(,u-l)lc2 _C1

212~4(2,u-1)2. 0

3. Coefficient bounds

For the larger class S· of starlike functions, R. Nevanlinna in 1920 [II, p. 46J

proved that the coefficient of each function f E S· satisfy Ian I~ n for n =2,3" . '. Brannan

et af. [2J obtained a sharp bound for the third coefficient of functions in SS * (a). We shall

give an alternate proof, and additionally, derive a sharp estimate for the fourth coefficient inthe result below. The general coefficient problem for the classes SS *(a) and PS *(p)

remains an open problem.

Theorem 1. Let fez) =z + a2z2 + a3z3 + ... E SS· (a). Then

la21~ 2a,with equality ifand only if

Further

Zf'(Z)=(I+&z)a, 1£1=1.fez) 1- &z

(9)

O<a::s;t

t::s;a::S;1

For a> 1/3, extremalfunctio"ns are given by (9). If 0 <a < 1/3, equality holds ifand only if

Zf'(Z)=(I+&z2)U,le l=l, (/0)fez) 1- &z2

while if a == I / 3, equality holds ifand only if

zf'(z) _ ()-a (1 1+ &z (I 1) 1- &z )-a---P2 z = 1\.--+ -I\. -- ,fez) I-&z 1+&z

Moreover,

I I j 2f, o<a::s;..[1;a4::S; 2: (l7a2 +1) M::s;a::S;1

For a ~ J2/17, extremalfunctions are given by (9), while for 0 < a ::s; J2/17, equality holds

ifand only if

zf'(z) =( I + &z3 JU , 1£ 1= I.fez) 1- &z3

179

Rosihan MAli

Proof The following relations are obtained from (3):

a2 =ac\

a[ 1-3a,,]a3 ="2 C2 --2-C\-

a [ 5a - 2 17a2-15a + 4 3] a E

a4 =- c3 +--clc2 + c\ :=-3 2 12 3

The bound on Ia2 I follows immediately from the well-known inequality Icl I~ 2. Lemma 2

with f.J =1- 3a yields the bound on I031 and the description of the extremal functions.

For the fourth coefficient, we shall apply Lemma 3 with 2/3 =(2 - 5a) / 2 and

<5 =(l7a 2 -15a + 4) /12. The conditions on /3 and <5 are satisfied if a ~ ~2/17. Thus

la41~2a/3, with equality ifand only if zf'(Z)/f(Z)=~I+a3)/(l-a3)r·In view of the fact that 0 < <5 < I, and <5 - /3 ~ 0 provided a ~ ~2/17, Corollary I

yields

lEI 17a2

- 15a + 4 I7a2

- 15a + 4 3 17a 2 - 21

II I 2 (17 2 I)~ c3 - cl c2 + ci + ci c2 S - a +

6 12 6 3

This completes the proof. D

Theorem 2. Let feSS*(a) alld f-l(w)=W+Y2W2 +Y3w3 + .... Theil

IY2 I~ 2a,with equality ifand only if

Z!'(Z)=(I+a)a,le l=1.f(z) I-a

Further

Ja, O<astIrJ\Slsa 2 , tSasl

For a > I /5, extremalfunctions are given by (9). If 0 < a < 1/5, eqilOlity holds ifand only if

zf'(z) =(1 + a2 )a, Iel= I,

f(z) 1- a 2

while if a =1/5, equality holds ifand only if

zf'(z) ( )-a _(.I+a (I .)I_a)-a--=P2 z - /L--+ -/L -- ,f(z) I-a I +a

180

The Coefficients ofFunctions

Moreover,

{

2a O<a<_I-< 3' -.J3i

Ir41- 2; (62a 2 +I} Jh'sa s I

For a ~ I /.f3I, extremal functions are given by (9), whilefor 0 <a s I/,J3I, equality holdsifand only if

Proof The following relations are obtained from (3) and (4):

(II)

As in the previous proof, the bounds on Ir2 I and IY3 I are obtained from the well-known

inequality Icl Is 2, and from Lemma 2.

For the fourth coefficient, we shall apply Lemma 3 with 2P = I + Sa and

0" =(31a 2 + 15a + 2)/6. The conditions on p and 8 are satisfied if as 1I.J3i. Thus

IY41s 2a /3, with equality ifand only if zf'(z)/ f(z)-:;: ~I + £z3)/(l- t:z3)f.

For 1/..[3i < a :$1/5, Corollary I yields

II I 1+5a 31 31a2

-11

13 21 2 )E S c3 - (l + 5a)c1C2 +--cI + cI :$-\62a + I2 6 3

It remains to determine the estimate for 1/5 < a :$ 1. Appealing to Lemma 4 with

p=5a, and because 31a 2 -15a+2>Oin (0,1], we conclude that

II I 31 31a2

-ISa+21

13 4 2E:$ c3 - (I + 5a)clc2 + Sac] + 6 CI s 2(10a -1) +3"(3Ia -ISa + 2)

= ~ (62a 2 + I) 03

Now we introducerespectively by,

zGn (z) -:;: (zn-I)Gn(z) q ,

the following functions in PS·(p). Define G",H,J e A

ZH'(Z)=q(z(z-r»), ZJ'(Z)=q(_z(z-r»), Osrsl.H(z) I - rz J(2.) I - rz

181

Rosihan M Ali

It is clear from (5) that Gn,H,JEPS*(p). Using (6), the following result can be

established in a similar fashion to Theorem 1.

Ib I< 16(1 - p)

2 - 2 '7r

with equality ifand only if g = G2 or its rotations. Further

{

8(1-P) (1. + 16(1-P»), 0 ~ p ~ 1- £I~I~ ;r2 3;r2 2 48

8(1-p) 1-L< 1;r2 ' 48 - P <

2 2For 0 ~ p < 1- ~8 ' equality holds if and only if g = G2 or its rotations. For 1- ~8 < P < I,

2

equality holds ifand only if g = G3 or its rotations. If p :: 1- ~8 ' equality holds ifand only

if g :: H or its rotations. Additionally,

{

16(I_P)[128(1-P)2 16(1-p) n] 0< < 1 £(1_ (89)2 4 + 2 +45' -P-+16 V45

Ib41~ 3;r ;r;r ,

16(1-p) 1+ '!!":"(I_ [89) < p < 13;r2 ' 16 v"45-

Equality holds in the upper expression oj the right inequality if and only if g:: G2 or its

rotations, while equality holds in the lower expression of the right inequality if and only ifg = G4 or its rotations.

4. Fekete-Szego Coefficient Functional

{

(S - 4t)a 2 , t ~ 5-~/a

Ir -tr 21< a 5-I/a <t< 5+I/a32- , 4--4

(4t - 5)a 2 , t ~ 5+~/a

If 5-~a < t < 5+~/a, equality holds if and only if J is given by (/0). If t < 5-~/a or

t> 5+~/a , equality holds ifand only if J is given by (9). IJ t = 5+~/a, equality holdr ifand

182

The Coefficients ofFunctions

if zf'(z) ( )aonly I fez) = P2 z ,

zf'(z) _ ()-afez) - P2 z .

while if t = 5-~a, then equality holds if and only if

Proof From (11), we obtain

2 a [ 1+ (5 - 4t)a 2]Y3- tY2 =-"2 c2- 2 c\.

The result now follows from Lemma 2. 0

Remark. An equivalent result for the Fekete-Szego coefficient functional over the classSS >I< (a) was also given by Ma and Minda [7].

2 3 >I<TheoremS. Lei g(z)=z+b2z +b3z +"'ePS (p). Then

2 2If 1_-"-- < t < 1 +-..2.L-, equality holds ifand only if g =G) or one ofits rotations. If

2 96(I-p) 2 96(1-p)

2 5 2t < t-96~-P) or t> t + 96(~-P)' equality holds if and only if g = G2 or one of its

2 .

rotations. If I = t - 96~ _p)' equality holds ifand only if g = H or one ofits rotations, while

2if t =-2

1 +-9Sf( ,then equality holds ifand only if g = J or one ofits rotations.6(1-p)

Finally, we note that the estimates above can be used to determine sharp upperbounds on the second and third coefficients respectively.

Acknowledgment. This research was supported by a Universiti Sains Malaysia FundamentalResearch Grant.

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Methods Function. Theory, World Scientific (1995), 23-36.[2] D.A. Brannan, 1. Clunie and W.E. Kirwan, Coefficient estimates for a class ofstarlike

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London Math. Soc. (2) t (1969),431-443.

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Rosihan M. Ali

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