*Proceedings ofthe SEAMS-GMU Conference 2003*

### The Coefficients of Functions with Positive Real Part and Some Special Classes of Univalent Functions

Rosihan M. Ali

School 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 on Mathematics and Its Applications was held on 14 -

17### July 2003 at Gadjah Mada University, Yogyakarta, Indonesia. The Conference is the forth conference held by Gadjah Mada University and SEAMS. The former was held in 1989, 1995 and 1999.

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

### During the 4-day conference there were

13### plenary lectures and

117### contributed 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. Suthep Suantai (Thailand), Prof. V. Dlab (Canada) and Dr. Widodo (Indonesia). Most of the contributed papers were delivered by Mathematicians from Asia.

### The proceedings consists of

5### invited lectures and

64### refereed contributed papers.

### In this occasion, we would like to express our gratitute and appreciation to the following 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, the participants and the referees for the wonderful cooperation, the great coordination and the fascinating effort. We would like to thank to our colleagues who help in editing papers especially to Atok Zuliyanto, Imam Sholekhudin, Fajar Adikusumo, I Gede Mujiyatna and Sri Haryatmi. Finally, we would like to acknowledge and express our thanks for the help and support of the staff and friends in the Mathematics Department, UGM in the preparation for and during the conference.

### Editorial Board Lina Aryati Supama Budi Surodjo Ch. Rini Indrati

iv

### CONTENTS

### Cover Page Preface Contents

**INVITED SPEAKER PAPERS**

### Optimization for Large Gas Transmission Network Using Least Difference Heuristic Algorithm

*Edy Soewono, Septoratno Siregar, Inna Savitri Widiasri and Nadia Ariani*

An ### Immersed Interface Technique for Mixed Boundary Value Problems *Gunther Peichl*

### Interaction between Mathematics and Finance *Hans van der Weide*

### On the Bounded Variation Interval Functions *Soeparna Darmawijaya*

### Topological Entropy of Discrete Dynamical System *Widodo*

**SELECTED CONTRIBUTED PAPERS**

1ll V

### 14

### 24

### 33

### 51

**Part I: Algebra**

### Convergence of the Divide-et-impera Algorithm in Case ofa Decoupled 63 System

*Ari Suparwanto*

### Finiteness Conditions of Generalized Power Series Rings 75 *Budi Surodjo*

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

*Caturiyati*

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

### On Cyclic N-ary Gray Codes 98

*I Nengah Suparta and A.* *1. van Zan ten*

v

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

### An Extremal Problem on the Number ofYertices of Minimum Degree in 112 Critically Connected Graphs

*Ketut Budayasa*

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

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

### Tabu Search Based Heuristics for the Degree Constrained Minimum Spanning 133 Tree Problem

*Wamiliana and Louis Caccetta*

### Necessary Conditions for The Existence of *(d, k)-Digraphs Containing* 141 Selfrepeats

*Y.M.* *Cholily and E. T. Baskoro*

**Part II: Analysis**

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

### Bilipschitz Triviality Does Not Imply Lipschtiz Regularity 155 *Dwi Juniati*

### Analytic Geometry in the Real Hilbert Space 160

*Imam Solekhudin and Soeparna Darmawijaya*

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

*Manuharawati*

### The Coefficients of Functions with Positive Real Part and Some Special Classes 173 of Univalent Functions

*Rosihan* *M.* *Ali*

(J' -

### Order Continuous and Ortogonally Additive Operators On a Space of 185

### Banach Lattice Valued Functions *Supama and Soeparna Darmawijaya*

VI

### Reserve-Cesaro-Orlicz Sequence Spaces and Their Second Duals *Y. M. Sri Daru Unoningsih*

### On the Henstock Integral of Functions 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 208 *Asep*

*K.*

*Supriatna and Edy Soewono*

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

### Modeling of Wood Drying 227

*Edi Cahyono. Tjang Daniel Chandra and La Gubu*

### The application of Implisit Kalman Filtering on A One Dimensional Shallow 234 Water Problem

*Erna Apriliani and A. W. Heemink*

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

### Waveform Relaxation Techniques for Stochastic Differential Equations with 248 Additive Noise

*Gatot F. Hertono*

### Higher Order Averaging in Linear Systems and an Application 256 *Hartono*

### Internal Solitary Waves in the Lombok Strait 265

*Jaharudin,* *S. R. Pudjaprasetya and Andonowati*

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

*Janson Naiborhu*

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

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

vii

### Perturbation of a Discrete Generalized Eigenvalue 303 *Lina Aryati*

### A Level Set Method for Two-Phase Steam Displacing Water in A Saturated 309 Porous Medium

*M. Muksar, E. Soewono and* S. *Siregar*

### The Effect of Vaccinations in a Dengue Fever Disease Transmission with Two- 321 strain Viruses

*Nuning Nuraini, Edy Soewono and Kuntjoro Adji Sidarto*

### Exsistence of Nash Solution for Non Zero-Sum Linear Quadratic Game With 331 Descriptor System

*Sa/mah,* S. *M* *Nababan, Bambang* S. *and* S. *Wahyuni*

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

*Sarwadi*

### A Finite Element Approach to Direct Current Resistivity Measurements in 349 Aeolotropic Media- The Forward Problem

*Sri Mardiyati, Peg Foo Siew and Yong Hong Wu*

### Characteristic of Two Wave Group Interactions of an Improved Kdv Equation 358 *Sutimin and*

*E.*

*Soewono*

### Some Criteria for Formal Complete Integrability of Nonlinear Evolution 366 Equations

*Teguh Bharata Adji*

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

### Feasible Direction Technique for Quadratic Programming Problems In 2 And 3 384 Dimension

*Yosza Dasril, Mustafa Mamat and Ismail bin Mohd.*

### Construction and Reconstruction of 3D Face Images Uses Merging and 391 Splitting Eigenspace Models

*Yudi Satria and Benyamin Kusumoputro*

VIII

### Part IV: Computers Sciences

### A Comparison Study of the Perfonnance of the Fourier Transfonn Based 400 Algorithm and the Artificial Neural Network Based Algorithm in Detecting

### Fabric Texture Defect

*Agus Harjoko, Sri Hartati and Dedi Trisnawarman*

### Development of 3D Authoring Tool Based on VRML 407

*Bachtiar Anwar*

### Right Inverse in Public Key Cryptosystem Design 414

*Budi Murtiyasa, Subanar, Retantyo Wardoyo and Sri Hartati*

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

### A study of High-Availability Disk Chace Manager for Distributed Shared- 428 Disks Using a Genetic Algorithm

*Pall-am, Han and Kwang-Youn, Jin*

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

### Iris Recognition System using Hadamard Transfonnation 446 *Lee Xuan Truong*

### Part V: Statistics

### Interval Estimation for Survivor Function on Two Parameters Exponential 452 Distribution under Multiple Type-II Cencoring on Simple Case with Bootstrrap

### Percentile

*Akhmad Fauzy, Noor Akma Ibrahim, Isa Daud and Mohd. Rizam Abu Bakar*

### The Stationary Conditions Of The Generalized Space-Time Autoregressive 460 Model

*Budi Nurani Ruclljana*

*Realibility Analysis of k-out-of-n Systems using Generalized Lambda* 466 Distributions

*Ha.\)'im Gautama and A.* 1. C. *van Gemund*

### Regularized Orthogonal Least Squares-based Radial Basis Function Networks 475 and Its Application on Time Series Prediction

*Hendri Murfi and Djati Kerami*

IX

### Analysis Of Relation Of Bernoulli Automorphism And Kolmogorov 484 Automorphism

*Henry Junus Wattimanela*

### Multivariate Outlier Detection using Mixture Model 489

*Juergen Franke and Bambang Susanto*

### Bandwith Choice for Kernel Density Estimation 494

*Kartiko*

### Statistical Properties of Indonesian Daily Stock Returns And Stochastic 500 Volality Models

*Khreshna Syuhada and Gopalan Nair*

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

### The Intervals Estimation for Intercept using Bootstrap Methods (Case Of 515 Bootstrap Persentile And BCA)

*Muhammad Safiih Lola and Norizan Mohamed*

### Bayesian Mixture Modeling in Reliability: MCMC Approaches 521 *Nur lriawan*

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

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

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

### Graphs

*Suhardi Djojoatmodjo*

### Note on Renewal Reward Processes 551

*Suyono and J A. M. van der Weide*

### The Estimation Parameters In Space Time Autoregressive Moving Average X 557 (STARMAX) Model For Missing Data

*Udjianna* S. *Pasaribu*

x

*Proceedings ofthe SEAMS-GMU Conference 2003*

### The Coefficients of Functions with Positive Real Part and Some Special Classes of Univalent Functions

Rosihan M. Ali

School 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.* Using
these estimates, we detennine sharp bounds for the first four coefficients over the classes
*SS·(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-Szego
coefficient functional is also treated.

*Keywords:*

*Univalent functions, analytic functions with positive real part, parabolic starlike*

*functions. 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 starlike functions 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 segment joining 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 by the 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 very good unified treatment of such a study under a weaker condition that the region is starlike with 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

### I

^{arg}

^{zf'(z)}^{f(z)}^{< -}

^{1fa}

^{2}

^{(z}

^{E}

^{U).}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: v*

*:::;*

^{2}### 4(1- *p)(u - p)}= {w:\ w-1\:::; 1- 2p* + Rew}.

*The class of parabolic 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 recall that 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)EPs*(t)*

If

is in the class SS

### *

*(a)*

*(or PS*

**(p»,*

then the inverse of*f*

admits an expansion
*f*

^{-1 (}

^{w)=w+Y2w}^{2}

^{+Y3w}^{3}

^{+ ...}

(I)

(2)

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

### I

*lover both classes SS*

^{an}### *

^{(a)}

^{and PS}### *

*(p), the first four coefficients of*

### I *y*

*n*lover 55

### *

*(a),*and find all possible extremal functions. Although the natural choice for an extremal function would arise from

*p(z)*= :~~

^{E}

*we show that it cannot be the sole extremal function for these problems. Additionally, we obtain sharp estimate for the Fekete- Szego coefficient functionals*

^{P,}*la3 - ta2*

### 21

or### Iy

^{3 -}

^{ty}^{2}

### 21·

174

*The Coefficients ofFunctions*

**2. Preliminary results**

The classes SS

### *

*(a)*and

*PS* *

*(p)*are closely related to the class

*P.*

Itis 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

### *

*and its inverses may be considered as non- linear coefficient problems for the class*

^{(a)}*P.*

Turning to the class

*PS·*

*Ali and Singh [I] has shown that the nonnalized Riemann mapping function*

^{(p),}*q*from

*U*onto

### n

*p*is given by

### () I 4(1-P)[1 1+..[;]2

*q*

z ### =

^{+}

^{o g - -}

1(2

### 1-..[;

*If f(z)*=z+

*b*

_{2}*z*

*+*

^{2}*b*

*+ ...E*

_{3}z^{3}*PS·*

*(p),*and

*h(z)*=

*zf'(z)/ f(z),*then there exists a Schwarzian function

*w*in

*U*with

*w(O)*=

### o. I

*w(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.* Itis 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 given in the following lemma.

Lemma 1 (5]. *Afunction p(z)*

### =

I+*Lk=1*

*Ck zk belongs to P*

*if*

*and 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 I*zk* III*k*<1.

### I

^{c2}^{--cl}

^{P}^{21}

::5:max{2,2Ip-lll=^{{}

^{2,}

### 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*

*Ie 1=I.*

^{2}),*For*

*,u*= 0,

*equality*

*holds*

*if*

*and only*

*if*

l+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*

*if*

*and 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

### I

^{C2 -}

_{~} ^{CI2r}

^{CI2r}

^{+}

^{I}

^{ci}

^{1}

^{2}

_{~} ^{1(1-}

^{f.J)cd}^{2}

^{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

### I

cl### 1=

0 and### I

c2### 1=

2, i.e.,*p(z)*

### =

(l### +

z2*)/0-*

*z2)*or its rotations. Equality holds when

*f.J*

### =

0 if and only jf### I

c2### 1=

2, i.e., [I I, p. 41]I+a I-a

*p(z):=P2(z)=,t--+(I-,t)--,* O~A.:::;I,

### 1&1=1.

I - a I+a

Finally, when *f.J*=2, then

### I

c2 - 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

Ic3 *-(fl+Y)Clc2 +ay'*

### '12

~4+4y(y-l)lc!### I

^{2}

*+(28-y)c]*

^{I}

### 2

*-(2fl-l)c2*

^{1}

^{2}-(2

^{,}

*--J'C\*

### 21 I

^{2}

I 1

2 2

2 *v* ^{2} ^{.} (8 - *fly)* ^{4}

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

### ICII

*2* *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 *fJ(2f3-1)~8~fJ,*

^{and}

^{then}### IC3 -

*2fJc(c2*+&13 /

### ~ ^{2.}

*Proof* If *{J*

### =

0, then 8### =

0 and the result follows since### I

c3### 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-p2*

*)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.}

Corollary1.

*If*

*p(z)=I+Lk'=lckZk eP, and*O~P~I,

^{then}### IC3 - *2f3c1*

c2 +### f3c1 3

1

### ~

^{2.}

*When P*

### =

0,*equality holds*

*if*

*and only*

*if*

3 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+cz3)/(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 I*C3* 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

^{c3 -}

^{2f3cIC2}^{+}

^{f3c1}^{312}

~^{4}+

*4fJ(fJ -*I)

### lei

1^{2}

*+4fJ(fJ -*I)

### I

*c2 --tCI*

^{21} ^{2}

*- P(P -*I)

### lei I

^{4}

~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 1*ci* 1= 2. If

### I

*ci*1=O. then 1

*c2*1= 0, i.e.,

*p(z)=(I+t:z*1&1=1. Iflcll=2, then

^{3})/(I_cz^{3}),*p(z)=(I+t:z)!(1-t:z),*1&1=1. 0 Lemma 4.

*If*

*p(z)*=I+

*Ik=rCkZk*E

*P, then*

### I

^{C}

^{3 -}

^{(p}^{+}

^{l)cl c}

^{2}

^{+}

^{f.lC131~}

max{2,212p -II}= { 2,
2/2p-l/,
O~p~1

*elsewhere*

*Proof* For 0~*p* ~I, the estimate follows from Lemma3with *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

*I*^{C}*3*-(,u+l)cl c2 +,uc1

### 31

^{2}

### ~4+4,u(,u-l)lc2

^{_C1}

^{2}

^{1}

^{2}

### ~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 in the result below. The general coefficient problem for the classes SS

### *

*and*

^{(a)}*PS*

### *

^{(p)}remains an open problem.

Theorem1. *Let fez)*

### =

z+*a2z2*+

*a3z3*+ ...

^{E}

*SS·(a).*

*Then*la21~

*2a,*

*with equality*

*if*

*and only*

*if*

*Further*

*Zf'(Z)=(I+&z)a,*

### 1£1=1.

*fez)* 1-*&z* (9)

### O<a::s;t t::s;a::S;1

*Fora>*1/3, *extremal functio"ns are given by*(9).

*If*

0### <

*a*

### <

1/3,*equality holds*

*if*

*and only*

*if*

*Zf'(Z)=(I+&z2)U,lel ^{=l,}*

*(/0)*

*fez)* 1-*&z2*
*while*

*if* *a*

==### I / 3,

*equality holds*

*if*

*and only*

*if*

*zf'(z) _* *()-a*

### (1

^{1+}

^{&z}^{(I}

### 1)

^{1-}

^{&z )-a}*- - - P 2* z

### = ^{1\.--+}

^{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,

*extremal functions are given by (9), while for*

### 0

<*a* ::s; J2/17,

*equality holds*

*if*

*and 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 -}

^{- - 2 -}

^{C\-}*a [* *5a -*

### 2 17

^{a}### 2- 15a

+### 4 3]

^{a E}*a4* = - c3 + - - c l c 2 + c\ :=-

3 2 12 3

The bound on

### I

*a2*

### I

follows immediately from the well-known inequality### I

cl I~2. Lemma 2 with*f.J*

### =

^{1-}

*3a yields the bound on*

### I

*031 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
### l *EI* ^{17}

^{a}^{2}

^{-}

^{15a}

^{+}

^{4}

^{I}

^{7}

^{a}^{2}

^{-}

^{15a}

^{+}

^{4}

^{3}

^{17}

^{a}^{2 -}

^{2} 1 II ^{I} 2 (17

2 I)
~ c3 - cl c2 + ci + ci c2 S -

*a* *+*

6

### 12

6 3This completes the proof. D

**Theorem**

### 2.

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

*Theil*

### I

*Y2*I~

*2a,*

*with equality*

*if*

*and only*

*if*

Z!'(Z)=(I+a)a,le_{f(z)}

### l

^{=1.}

I-a
*Further*

### J ^{a,} O<ast IrJ\Slsa

^{a,}

^{2 ,}

### t Sasl

*For a*>I

### /5,

*extremal functions are given by*(9).

*If*

0<*a*<

### 1/5,

*eqilOlity holds*

*if*

*and only*

*if*

*zf'(z)*

### =(1 +

*a*

^{2}*)a,* ^{Iel=}

^{I,}

*f(z)* 1- a ^{2}
*while*

*if*

*a*

### = 1/5,

*equality holds*

*if*

*and 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}

^{(62a}

^{2}^{+} ^{I}} ^{Jh'sa} ^{s I}

^{Jh'sa}

*For a*~

### I /.f3I,

*extremal functions are given by*

### (9),

*while for*

### 0

<*a*

### s I/,J3I,

*equality holds*

*if*

*and only*

*if*

*Proof*

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

As in the previous proof, the bounds on

### I

*r2*

### I

and### I

*Y3*

### I

are obtained from the well-known inequality### I

*cl*

### 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
### I

*Y41s 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

^{E}^{S} I

^{c3 -}^{(l}

^{+}

^{5a)c1C2}^{+}

^{- - c I}^{1+5a}

^{2} ^{31}

^{+}

^{31a}

^{6}

^{2}

^{-1}

^{1}

^{cI}^{1} ^{3}

^{:$-\62a}^{21} ^{3}

^{21}

^{2}

^{+}

^{I}

^{)}

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 ^{E:$} I

^{E:$}

^{c3 -}^{(I}

^{+}

^{5a)clc2}^{+}

^{Sac]}^{31} ^{+}

^{31a}

^{2}

^{-ISa+2}

^{6}

^{1}

^{CI}

^{1}

^{3}

^{s}

^{2(10a}

^{-1)}+3"(3Ia

^{4}

^{2}-ISa

### +

2)=

### ~ *(62a*

^{2}+I) 0 3

Now we introduce respectively 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.

### I ^{b}

^{b}

_{2 -}

### I

^{<}

^{16(1 -}

_{2}

^{p)}_{'}

7r

*with equality*

*if*

*and only*

*if*

g= G_{2}

*or its rotations. Further*

{

*8(1-P)*

### (1.

^{+}

^{16(1- P}^{»),}

^{»),}

^{0}

### ~

^{p}### ~

^{1-}

### £

I~I~ ^{;r2} ^{3 ; r 2} ^{2} ^{48}

*8(1-p)* *1-L<* *1*

;r2 ' 48 - *P<*

2 2

*For* 0_{~} *p*

### <

1-_{~8}'

*equality holds*

*if*

*and only*

*if*

g=G_{2}

*or its rotations. For*1-

_{~8}

### <

*P*<I,

2

*equality holds*

*if*

*and only*

*if*

^{g}=G

_{3}

*or its rotations.*

*If*

*p ::*1-~8'

*equality holds*

*if*

*and only*

*if*

g ::*H or its rotations. Additionally,*

{

16(I_P)[128(1-P)2_{2} _{4} _{+}*16(1-p)*_{2} _{+45'}

*n]* ^{0}

^{<}- P - + 1 6

^{<}

^{1 £(1_} V45 ^{(89)}

Ib41~ ^{3;r} ^{; r ; r} ^{,}

*16(1-p)*

### 1 + '!!":"(I_ [89)

^{<}

^{p}^{<}

^{1}

3;r2 ' 16

### v"45-

*Equality holds in the upper expression oj the right inequality*

*if*

*and only*

*if*

g::G_{2}

*or its*

*rotations, while equality holds in the lower expression of the right inequality*

*if*

*and only*

*if*

g = G4 *or its rotations.*

**4. Fekete-Szego Coefficient Functional**

*{*

*(S - 4t)a2 ,* *t*

### ~ 5-~/a

### Ir

^{3 2 -}

^{-tr} ^{21<} ^{a}

^{-tr}

^{a}

^{,}

^{5-I/a}

^{4 - - 4}

^{<t<}^{5+I/a}

*(4t - 5)a2 ,* *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*

*if*

^{and only}*if*

*J is givenby*(9).

*IJ t*=5+~/a,

*equality holdr*

*if*

^{and}182

*The Coefficients of Functions*

*if*

^{zf'(z)}^{(}

^{)a}*only* I *fez)* =*P2* *z* *,*

*zf'(z) _* *()-a*

*fez) -* *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-*^{t}*Y2*

### =-"2

*c2-*2

*c \ .*The result now follows from Lemma 2. 0

Remark. An equivalent result for the Fekete-Szego coefficient functional over the class

SS^{>I<}*(a)* was also given by Ma and Minda [7].

2 3 ^{>I<}

TheoremS. *Lei g(z)=z+b _{2}z +b3z +"'ePS (p). Then*

2 2

*If* 1_ - " - -

<*t*<

### 1

+-..2.L-,*equality holds*

*if*

*and only*

*if*

g ### = G)

*or one of its rotations.*

*If*

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

2 5 2

*t*<

### t- 96~-P)

^{or t>}### t

^{+}

### 96(~-P)'

*equality holds*

*if*

*and only*

*if* g

=### G

2*or one of its*

2 .

*rotations.*

*If*

*I*=

### t -

^{96~}

^{_}

^{p)'}*equality holds*

*if*

*and only*

*if* g

=*H or one ofits rotations, while*

2

*if t*

### =

^{-2}

^{1}+- 9

*Sf(*

*,then equality holds*

*if*

*and only*

*if*

g=J *or one of its rotations.*

6(1-p)

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

Acknowledgment. This research was supportedbya Universiti Sains Malaysia Fundamental Research Grant.

**References**

[1] R.M. Ali and V. Singh, *Coefficients of parabolic starlike functions of order p,* Compo
Methods Function. Theory, World Scientific (1995), 23-36.

[2] D.A. Brannan, 1. Clunie and W.E. Kirwan, *Coefficient estimates for a class of starlike*
*functions,* Canad.J. Math. 22(1970), 476-485.

[3] D.A. Brannan and W.E. Kirwan, *On some classes of bounded univalent functions,* J.

London Math. Soc. (2)

### t

(1969),431-443.183

*Rosihan* *M.* *Ali*

[4] A.W. Goodman, *On uniformly convex functions,* Ann. Polon. Math. 56 (1991), 87-92.

[5] C.R. Leverenz, *Hermitian forms in function theory,*Trans. Amer. Math. Soc. 286 (1984),

### 675-688.

[6J R.J. Libera andEJ. Zlotkiewicz, *Early coefficients of the inverse of a regular convex*
*function,* Proc. Amer. Math. Soc.

### 85

(1982), 225-230.### [7J

W. Ma and### D.

Minda,*An internal geometric characterization of strongly starlike*

*functions,*Ann. Univ. Mariae Curie-Sklodowska Sect. A45(1991), 89-97.

[8J W. Ma and D. Minda,*A unified treatment ofsome special classes of univalent functions,*
Proc. Conf. on Complex Analysis, Tianjin (1992), 157-169.

[9] W. Ma and D. Minda, *Uniformly convex functions,* Ann. Polon. Math 57 (1992), 165-
175.

[10]M. Nunokawa and S. Owa,*On certain conditions for starlikeness,*Southeast Asian Bull.

Math. 25 (2001),491-494.

[II] Ch. Pommerenke, *Univalentfimctions.* Vandenhoeck and Ruprecht, Gottingen 1975.

[12] F. R0nning, *Uniformly convex fimctions and a corresponding class ofstarlike functions,*
Proc. Amer. Math. Soc. 118 (1993), 189-196.

[13JJ. Stankiewicz,*Some remarks concemi*^{1}*!g starlike fil1lctions,* Bull. Acad. Polon. Sci. Ser.

Sci. Math. Astronom. Phys. 18 (1970), 143-146.

[14]1. Stankiewicz, *On a family of starlike functions,* Ann. Univ. Mariae Curie-Sklodowska
Sect. A 22-24 (1968/70),175-181.

184