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.

Tekspenuh

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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

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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

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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

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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

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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

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

On the Henstock Integral of Functions Defined on

Rn

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

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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

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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

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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

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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 ordera 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:1z

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

EA 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 WEfeU) 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 Rep(z)>0, ZE

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

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Rosihan M. Ali

An analytic function

f

EA is said to be strongly starlike of order a, 0<a:::;I, if it satisfies

I

argzf'(z)f(z) < -1fa2 (z EU).

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

Qp

= {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 , zEU. 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 ifzf'(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+Y2w2+Y3w3+ ...

(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

anlover both classes SS

*

(a) and PS

*

(p), the first four coefficients of

I y

nlover 55

*

(a), and find all possible extremal functions. Although the natural choice for an extremal function would arise from p(z)= :~~EP, we show that it cannot be the sole extremal function for these problems. Additionally, we obtain sharp estimate for the Fekete- Szego coefficient functionalsla3 - ta2

21

or

Iy

3 - ty2

21·

174

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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

*

(a) and its inverses may be considered as non- linear 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+..[;]2

q

z

=

+ o g - -

1(2

1-..[;

If f(z)=z+

b

2

z

2+

b

3z3+ ...E

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

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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 Izk IIIk<1.

I

c2 --clP

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. If0<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

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

(14)

The Coefficients ofFunctions

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

=

I, and zk

=

0 ifk>I.This yields

I

C2 -

~ CI2r

+Ici 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

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 ofP2' 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 ifk>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)c212-(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 andfJ(2f3-1)~8~fJ,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

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

p is the reciprocal ofP3' 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)lcI14 ~ 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

I

ci 1=O. then 1c21= 0, i.e., p(z)=(I+t:z3)/(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 EP, then

I

C3 - (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 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

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The Coefficients ofFunctions

from (7) and (8) that

IC3-(,u+l)cl c2 +,uc1

31

2

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

_C12

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

ES· satisfyIan

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

*

(a) and PS

*

(p)

remains an open problem.

Theorem1. Let fez)

=

z+a2z2 +a3z3 + ...ESS·(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\.--+

- 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

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Rosihan MAli

Proof The following relations are obtained from (3):

a2 =ac\

a[ 1-3a,,]

a3

="2

C2

- - 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

=

(l7a2

-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

a2-

15a

+4 I

7

a2-

15a

+4 3

17

a2 -

2 1 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

I

Y2 I~

2a,

with equality

if

and only

if

Z!'(Z)=(I+a)a,lef(z)

l

=1.

I-a Further

J a, O<ast IrJ\Slsa

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

(18)

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),

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"

=

(31a2+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 I1+5a

2 31

+31a

6

2-1

1

cI

1 3

:$-\62a

21 3

2 +

I

)

It remains to determine the estimate for 1/5<

a

:$ 1. Appealing to Lemma 4 with p=5a, and because 31a2-15a+2>Oin (0,1], we conclude that

II E:$ I

c3 - (I+5a)clc2 +Sac]

31 +

31a2-ISa+26 1CI13

s

2(10a-1)+3"(3Ia4 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

(19)

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

2 -

I

<

16(1 -

2p)'

7r

with equality

if

and only

if

g= G2 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=G2 or its rotations. For 1-~8

<

P<I,

2

equality holds

if

and only

if

g=G3 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)22 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::G2 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

, 5-I/a4 - - 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

(20)

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

ProofFrom (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 class

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

2 3 >I<

TheoremS. Lei g(z)=z+b2z +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

=

-21+- 9Sf( ,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

(21)

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 concemi1!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.

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