prr.hec.gov.pkprr.hec.gov.pk/jspui/bitstream/123456789/1624/1/2140S.pdf · iii On Certain Subclasses of Analytic Functions in Geometric Function Theory of a Complex Variable ____________________________________________

On Certain Subclasses of Analytic Functions inGeometric Function Theory of a Complex Variable

By

Kamran Yousaf

CIIT/FA09-PMT-001/ISB

PhD Thesis

In

Mathematics

COMSATS Institute of Information Technology

Islamabad, Pakistan

Spring, 2013

ii

COMSATS Institute of Information Technology


A Thesis Presented to

COMSATS Institute of Information Technology, IslamabadIn partial fulfillment

Of the requirement for the degree of

PhD Mathematics

By

Kamran Yousaf


Spring, 2013

iii

On Certain Subclasses of Analytic Functions inGeometric Function Theory of a Complex Variable____________________________________________

A Post Graduate Thesis submitted to the Department of Mathematics as partial

fulfillment for the award of Degree of Ph.D Mathematics.

Kamran Yousaf CIIT/FA09-PMT-001/ISB

Supervisor

Prof. Dr. Khalida Inayat Noor

Professor

Department of Mathematics

COMSATS Institute of Information Technology (CIIT), Islamabad Campus

June, 2013

Name Registration No.

iv

Final Approval

This thesis titled


By

Kamran Yousaf

CIIT/FA09-PMT-001/ISBHas been approved

For the COMSATS Institute of Information Technology, Islamabad

External Examiner 1:__________________________....................................................................................................

External Examiner 2:__________________________....................................................................................................

Supervisor:________________________Prof. Dr. Khalida Inayat NoorDepartment of Mathematics

HoD:_________________________Prof. Dr. Moiz Ud Din KhanDepartment of Mathematics

Chair Person:________________________Prof. Dr. Tahira Haroon

Dean, Faculty of Sciences:______________Prof. Dr. Arshad Saleem Bhatti

v

Declaration

I, Kamran Yousaf, registration number CIIT/FA09-PMT-001/ISB, hereby declare that I

have produced the work presented in this thesis during the scheduled period of study. I also

declare that I have not taken any material from any source except referred to wherever due,

that amount of plagiarism is within acceptable range. If a violation of HEC rules on research

has occurred in this thesis, I shall be liable to punishable action under the plagiarism rules of

the HEC.

Signature of student:_____________

Date:_________ Kamran Yousaf


vi

Certificate

It is certified that Kamran Yousaf bearing CIIT/FA09-PMT-001/ISB has carried out all the

work related to this thesis under my supervision at the Department of Mathematics

COMSATS Institute of Information Technology Islamabad and the work fulfills the

requirement for award of PhD degree.

Date:____________

Supervisor

_______________________

Prof. Dr. Khalida Inayat Noor

Professor


Head of Department

______________________

Prof. Dr. Moiz Ud Din Khan

Professor


vii

DEDICATION

This work is dedicated

to

my sisters Qudsia and Shahida

whose

prayers are assets of my life

and made me what I am today.

viii

ACKNOWLEDGEMENTS

In the name of Almighty Allah, the most Gracious and the most Merciful, all praise

and glory to Allah for His blessings upon me to finish my thesis. I glorify His name for

giving me the strength and the courage in fulfilling this requirement. All respect to our Holy

Prophet Muhammad (peace be upon him) who enables us to recognize Allah and who is a

source of knowledge and guidance for the whole mankind.

I would also like to show my gratitude and greatest appreciation to my supervisor

Prof. Dr. Khalida Inayat Noor, (Pride of Performance) for giving me the guidance and her

time as well as permanent orientation and encouragement which are beyond description and

words during the course of my study. Her contribution cannot be acknowledged in a few

lines packed with gratitude.

I am highly obliged and very grateful to Dr. S.M. Junaid Zaidi, Rector COMSATS

Institute of Information Technology, for providing each type of latest research facility and

for creating conducive environment in the institute. Special thanks for the Dean, Faculty of

Sciences Prof. Dr. Arshad Saleem Bhatti (Tamga-e-Imtiaz) and the faculty members, for

extending the much required administrative support.

I have strong feelings of appreciation for the Higher Education Commission, Pakistan for

financial support through indigenous 5000 Ph.D. fellowship scheme, Batch-IV and for

providing me the latest literature in the form of the most updated digital and reference

libaries.

My special thanks goes to Honorable Prof. Dr. Muhammad Aslam Noor

Department of Mathematics, for his invaluable suggestions . I am also grateful to the Head

Department of Mathematics CIIT Islamabad, Prof. Dr. Moiz ud Din Khan, for providing us

with research facilities in the Department of Mathematics.

Of course I am very thankful to my brother and sisters for their love and affection. This study

would have been impossible without the prayers, love, help, encouragement and moral

support of my family.

My thanks are also for all my colleagues and friends for their help and understanding.

ix

My special thanks to my parent institution Government Degree College, Lassan Nawab,

Mansehra for providing me the opportunity for higher education.

Kamran Yousaf

CIIT / FA09-PMT-001 / ISB

x

ABSTRACT


In this thesis, we define some new subclasses of analytic functions using the techniques of

convolution, differential subordination and the concepts of conic domains, bounded boundary

rotation and bounded radius rotation. We study these new classes thoroughly and investigate

several coefficient results, radii problems, convolution preserving properties, inclusion

results, integral preserving mapping properties along with various other useful applications.

Most of these results are sharp. Various known and new results are also derived as special

cases from our results.

xi

TABLE OF CONTENTS

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Some Preliminary Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . .09

2.1 Some Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 The Carathéodory Class of Functions with Positive Real Part and Some

Related Classes . . .. . . . . .. . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Certain Subclasses of Univalent Functions . . . . . . . . . . . . . . . . . . . . .20

2.4 Functions with Bounded Radius Rotation and Bounded Boundary

Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30

2.5 Uniformly convex, Uniformly starlike and Related Functions. . . . .. . .33

2.6 Bazilevič Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37

2.7 Certain Linear Operators Defined in terms of Convolution . . . . . .. . . 38

2.8 Some Preliminary Lemmas.…… . . . . . . . . . . . . .. .. . .. . .. .. .. .. . . . . .41

3 Generalized Janowski Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 47

3.2 Main Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.3 Integral Operators And Radii Problems . . . . . . . . . . . . . . . . . . . . . . . .61

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .77

4 Bazilevič Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 79

4.1 Introduction . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .80

4.2 Main Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..88

5 Generalized Bazilevič Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . 92

xii

5.2 Preliminary Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 94

5.3 Main Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 94

5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .104

6 Non-Bazilevič Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .106

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.2 Main Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

7 Conclusion . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

8 References . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

xiii

LIST OF FIGURES

Fig 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 28

Fig 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...35

Fig 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .78

Fig 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .90

Fig 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105

Fig 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .118

Fig 7.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

xiv

LIST OF TABLES

NIL

xv

LIST OF ABBREVIATIONS

Set of complex numbers

Set of real numbers

E Open unit disk

2 1 , ; ;F a b c z Gauss Hypergeometric functions

k z Koebe function 0L z Möbius function

nx Pochhammer symbol

f g Convolution of f and g

Subordination

A Class of normalized analytic functions

S Class of univalent functions

P Class of Carathéodory functions

C Class of Convex functions

( )C Class of Convex functions of order λ

C Class of Convex functions of complex order

*S Class of Starlike functions

( ) Class of Starlike functions of order λ*S Class of Starlike functions of complex order

K Class of Close to Convex functions

( , )K Class of Close to Convex functions of order λ type µ*C Class of Quasi convex function*( , )C Class of Quasi convex function of order λ type µ

mR Class of analytic functions with bounded radius rotation

mV Class of analytic functions with bounded boundary rotation

( , , , )B g h Class of Bazilevič functions

( )N Class of Non-Bazilevič functions

xvi

UCV Class of uniformly convex functions

UST Class of uniformly starlike functions

[ , ]P A B Class of Janowski functions

[ , ]C A B Class of Janowski Convex functions

*[ , ]S A B Class of Janowski Starlike functions

k Conic domain

nD Ruscheweyh derivative

cI Bernardi operator

nI Noor integral operator

( , )L a c Carlson - Shaffer operator

Chapter 1

Introduction

1

Introduction

Geometric Function Theory is the study of geometric properties of analytic functions.

The theory of analytic functions and more specic univalent functions have a long history

and remains a dynamic eld of research. The rst stirring of function theory was found

in the 18th century with the works of Euler. But still new applications arise continually.

In 1851, Riemann mapping theorem gave rise to the birth of Geometric Function Theory

and allowed mathematicians to reduce problems about simply connected domain D � C

to the particular case of open unit disk E = fz : jzj < 1g ; see [11, 15]. This theorem is

considered as one of the most fundamental contributions of Geometric Function Theory.

It should be noted that Riemann mapping theorem demonstrates the existence of a

mapping function but it does not exhibit this function.

Modern Function Theory was developed in the 19th century. The pioneers of Geomet-

ric Functions Theory were Euler, Weierstrass, Cauchy, Riemann, and several more. Each

gave the theory a very distinctive essence. Geometric Function Theory is mainly con-

cerned with the class S of univalent functions dened in E which was rst considered by

Koebe in 1907. The origin of univalent function theory can be traced to the 1907 paper of

Koebe, to Gronwalls proof of Area theorem in 1914 and to Bieberbachs estimate for the

second coe¢ cient of normalized univalent functions in 1916 and its consequences. Koebe

discovered that the functions which are both analytic and univalent in a simply connected

domain D � C have some nice geometric properties. The univalent function theory is

much complicated but fascinating area of research. Although Geometric Function Theory

is a classical subject, yet it continues to nd new applications in an ever-growing vari-

ety of areas such as modern mathematical physics, engineering, electronics, medicines,

uid dynamics, non-linear integrable system, theory of partial di¤erential equations and

many other branches of applied sciences. Many renowned mathematicians like Brannan,

Bieberbach, Koebe, Löewner, Miller, Mocanu, Noor, Paatero, Pommerenke, and several

2

others contributed in the development of Geometric Function Theory and gave it a new

distinctive character and dimensions, see [6, 11, 15, 37, 39, 49, 66, 71].

A close linkage of univalent functions with conformal mappings gave the subject a

gigantic potential of research and the growing applications of the subject arose in other

elds of sciences. In order to explore the geometric properties of analytic functions,

the major tools such as subordination and convolution have been recently used in this

area. The set of complex numbers and then complex functions are not ordered elds,

so the concept of subordination instead of inequalities was developed, for details, see

[39]. Furthermore, de-Branges [10] used the hypergeometric functions for proving the

Bieberbach conjecture and this fascinating application of hypergeometric functions in-

cited researchers to involve special functions in Geometric Function Theory. The research

in this area, on broad spectrum mostly deals with the geometric properties of univalent,

multivalent, meromorphic functions etc. and the theorists paid much of their attention

to the classes of these functions. In the course of tackling Bieberbach Conjecture,

new classes of analytic and univalent functions such as classes of convex and starlike

functions etc. [11, 15] were dened and some delicate properties of these classes were

widely investigated. These classes related with the Carathéodory functions were studied

systematically in various aspects. It was also observed that the class S of univalent func-

tions had connections with the classes of convex and starlike functions with some subtle

geometric properties.

A domain is star-shaped with respect to point w0 if the line segment joining w0 to

every other point of the domain lies entirely in that domain. A function f(z) is starlike if

it maps E onto a starshaped domain with respect to w0 = 0: Similarly a domain is convex

if it is star-shaped with respect to each of its point. A function f(z) is convex if it maps E

onto a convex domain. It was noted that both of these classes are contained in the class S;

for detail see [15]. Later Alexandar, Nevanlinna and many others extensively investigated

these classes. Moreover, the classes of convex and starlike functions are closely related

with the Caratheodory class P of analytic functions p(z) with p(0) = 1 and Re p(z) > 0,

3

see [11, 15]. In course of nding di¤erent classes of univalent functions, Kaplan [18]

introduced the class K of close-to-convex functions which has simple geometric meaning.

Geometrically, f(z) 2 K means that f(z) maps each circle jzj = r < 1 into a simple

closed curve whose tangent rotates, as � increases, either in a clockwise or anticlockwise

direction in such a way that it never turns back onto itself as much as to completely

reverse its direction. Then a new subclass of S known as class of quasi convex functions

denoted by C� was introduced and studied, see [49, 60]. It has the same relation with

the class K of close-to-convex functions as C has with S�:

A natural extension of the class of convex functions is the class Vm; m � 2. The

class Vm; m � 2 of functions with bounded boundary rotations was rst introduced by

Löewner [32] in 1917. Many renowed mathematicians like Brannan [6], Brannan et.al

[7], Noor [45, 54, 56, 58], Paatero [66] and several other investigated various aspects and

applications of the class Vm; m � 2: A function f (z) 2 Vm if it maps E conformally onto

a domain whose boundary rotation is at most k�. It is obvious that V2 = C, the class

of normalized convex functions. It is known [15] that, for 2 � m � 4, Vm � K � S. In

the same spirit Tammi [91] in introduced the class Rm of functions with bounded radius

rotations by extending the idea of starlike functions.

Later, in 1955, Bazileviµc [3] dened and studied the class of Bazileviµc functions, which

is the largest known subclass of univalent functions till now. A function f(z) 2 A is said

to be in the class of Bazileviµc functions if it satises the relation

f (z) =

8 0,

� is any real number and � > 0. A subclass of those functions for which � = 0 is also a

topic amongst heavily researched classes. That is, for � = 0 in (1) and on di¤erentiating,

we have

zf 0 (z) = (f (z))1�� g� (z)h (z) : (2)

4

The functions f(z) satisfying (2) are called Bazileviµc functions of type � . For � = 1, it

is observed that this class coincides with the class K of close-to-convex functions. It is

still unclear whether any geometrical meaning can be derived from (2) for other values of

�. Also the famous Beiberbach conjecture for Bazileviµc functions is still unsettled. Some

contribution in this regard is made by Zamorski [99]. In 1968, Thomas [92] tried to give

geometric characterization of these functions for � = 0. Sheil-Small [87] gave a natural

characterization of the ordinary Bazileviµc functions along the lines of Kaplan [18].

The notion of the class of Non-Bazileviµc functions N( � ) was rst introduced by

Obradovic [65] in 1998. Until now, this class has been studied in a direction of nding

necessary conditions over � that embeds this class into the class of univalent functions or

its subclasses. In recent years a large number of papers have appeared in the literature

concerned with extending the results contained in Obradovics paper [65]. Tuneski and

Darus [93] obtained Fekete-Szego inequality for the class of Non-Bazileviµc functions.

Using this concept of Non-Bazileviµcness, Wang et al [94] generalized this class and studied

its properties.

If f(z) and g(z) are analytic in E, we say that f(z) is subordinate to g(z); written as

f(z) � g(z), if there exists a Schwarz function w(z) in E such that f(z) = g(w(z)). By

using the principle of subordination, Janowski [17] introduced the class P [A;B] dened

as:

Let A and B be any two xed numbers with �1 � B < A � 1. Then P [A;B] is the class

of all functions p(z) analytic in E and satisfying the property

p (z) � 1 + Az1 +Bz

; z 2 E:

Brown [8] showed that it is not always true that f (z) 2 S� maps each disk jz � z0j < � <

1� jz0j onto a domain starlike with respect to f (z0) : He proved that the result is true for

each f (z) 2 S and for su¢ ciently small disk in E: Later, it is well known that, for any

complex-valued function f(z), not only f(E) but also the images of all circles centered at

origin and lying in E are convex arcs. Pinchuk posed a question whether this property is

5

still valid for circles centered at other points. Goodman [13, 14] gave a negative answer to

this question. This motivated the denition of uniformly starlike functions, though it was

also introduced independently in the work of Brown [8], Goodman [13] introduced the

class of uniformly convex functions. Later Ronning [81] Ma and Minda [35] and several

other obtained a suitable form of Goodmans criteria of uniformly convex functions, which

are related to conic regions. A function f (z) 2 S is uniformly convex ( starlike) if for

every circular arc � contained in E with center z0 2 E the image arc f (�) is convex (

starlike). The classes of uniformly convex and uniformly starlike functions are denoted

by UCV and UST respectively. The following two-variable analytic characterization of

the class UCV is important for obtaining information about functions in this class.

A function f (z) belongs to UCV if and only if

Re

�1 + (z � z0)

f 00(z)

f 0(z)

�� 0; z; z0 2 E: (3)

It is evident that UCV � C: However, by taking z0 = �z in (3), it is clear that UCV �

C�12

�: Also if we take z0 = 0; we obtain well known class C of convex functions [14]. A

function f (z) belongs to UST if and only if

Re

�(z � z0) f 0 (z)f (z)� f (z0)

�� 0; z; z0 2 E: (4)

Note that, by taking z0 = 0 in (3) and (4) above, we obtain the classes C and S�

respectively. The classic Alexanders theorem stating that f(z) 2 C if and only if

zf0(z) 2 S� , provides a bridge between these two classes. One might hope that there

would be a similar bridge between UCV and UST but the Alexander type result f(z) 2

UCV if and only if zf 0(z) 2 UST does not hold. The class

Sp = ST = fg (z) : g (z) = zf 0(z) where f (z) 2 UCV g ;

was introduced by Ronning [81] to verify whether Sp is a proper subclass of UST or not.

6

It turns out, see [13, 78], that there is no inclusion between UST and Sp: That is

UST * Sp and Sp * UST:

Ronning [81] and Ma and Minda independently have given a more applicable one variable

characterization of the class UCV , which is stated as below.

Let f(z) 2 A then f(z) 2 UCV if, and only if,

Re

�(zf 0(z))0

f 0 (z)

�>

��(zf 0(z))0f 0 (z) � 1�� ; z 2 E:

Similarly, Ronning [79] showed that an analytic function f(z) 2 ST if, and only if,

Re

�zf 0(z)

f (z)

�>

��zf 0(z)f (z) � 1�� ; z 2 E:

With the same analogy, Ronning [81] and several others introduced the classes of starlike

functions and its generalizations associated with conic regions.

In this thesis, we use the concepts of di¤erential subordination, the convolution di¤eren-

tial operator and conic domains, to dene various new classes of analytic functions in E.

Study of some subclasses of Bazileviµc and Non-Bazileviµc functions will be the prime ob-

jective of this thesis. For relevant references, see [1-97]. We shall investigate some basic

properties of these classes such as inclusion relations, integral preserving properties, ra-

dius problems, growth of coe¢ cients, convolution properties and some other challenging

problems. A brief introduction of all chapters is given below.

Chapter 2 emphasizes on some preliminary and classical concepts of Geometric

Function Theory which supply an essential environment for the investigation of the work

presented in the subsequent chapters. The proofs of the results in this chapter are omitted

but the relevant references are given. The two major tools subordination and convolution

are concisely discussed here. Some linear operators which provide a suitable framework

for the analysis of certain analytic functions are also discussed in this chapter. Some

7

lemmas, which are necessary to prove our main results in other chapters, are included.

We also discuss the classes of Bazileviµc and Non-Bazileviµc functions which are the main

focus of our thesis.

In Chapter 3 we employ classes, mentioned in the chapter rst, to dene a new class

Pm [A;B; �] for 0 � � < 1 and m � 2 of analytic functions related with the generalized

Janowski functions. All the contents of this chapter have already been published in

[62]. Classes Rm [A;B; �] and Vm [A;B; �] are also introduced in this chapter. we also

derive the necessary and su¢ cient condition for the function f (z) to belong to the class

Rm [A;B; �] : Some results such as inclusion result, a radius problem, coe¢ cients bounds

and study of integral operators for this class are investigated. Further we derive arc length

problem. For di¤erent choices of parameters we present the relationship of previously

known classes with this class.

Chapter 4 contains a new class k � UBm (�; �) for �; � 2 R; m � 2; and k 2

[0;1). The contents of this chapter have been accepted in the Journal of Computational

Analysis and Application. We also focus on some interesting properties of the subclass

k�UBm (�; �). The most interesting one is that the class k�UBm (�; �) is closed under

convex convolution. We have also proved that this class is invariant under some integral

representations.

Chapter 5 is devoted to dene new subclasses of analytic functions. We use classes

mentioned in the chapter rst to dene the class k � UBm(�; �; h; g) of generalized

Bazileviµc functions related with conic regions. The contents of this chapter have been

published, see [63]. We also focus on some interesting results related to this class.

In chapter 6 we dene the class N�;� (A;B) for � � 0, �1 � B < A � 1 and

0 < � < 1 by using the concept of di¤erential subordination, given by Miller and Mocanu

in [38]. Interesting properties of this class, such as a covering theorem, coe¢ cient bounds

and Fekete-Szego inequality are studied. We also investigate inclusion relationship. The

contents of this chapter have been published, see [73]. In the end, we include all the

relevant references used in this thesis.

8

Chapter 2

Some Preliminary Concepts

9

This chapter presents some signicant denitions and results of well-known classes of

analytic functions. This comprehensive survey takes a supportive part of the upcoming

chapters.

2.1 Some Basic Denitions

Due to the vital role of analytic functions in Geometric Function Theory, we shall discuss

analytic functions which are univalent in the open unit disk E = fz : jzj < 1g. The main

goal is to introduce the class of normalized univalent functions and some of its interesting

properties. For references of this section, see [11, 15]:

Denition 2.1.1 A complex valued function f (z) is said to be analytic at a point z0 if

its derivative exists not only at z0 but in some neighborhood of z0: A function f (z) is

called analytic in a domain D if it is analytic at each point in D: The exponential, the

sine and cosine functions are analytic in the whole complex plane.

Sometimes, tackling problems by considering an arbitrary domain D is an arduous and

gruelling task. For convenience, we restrict D to the open unit disk E. This replacement

is possible due to undermentioned Riemann mapping theorem.

Denition 2.1.2 Let D be a simply connected domain properly contained in C with

atleast two boundary points. Then there exists a unique analytic function which maps

D conformally onto the open unit disk E and has the properties f (z0) = 0 and f0(z0) >

0: This theorem demonstrates the existence of a function but it does not exhibit this

function.

Here we shall deal the class A of analytic functions in the open unit disk E normalized

by f (0) = 0 = f0(0)� 1: This normalization just eliminate unnecessary parameters and

does not a¤ect generality.

As every complex valued analytic function f (z) can be represented by power series, so

10

each f (z) 2 A has a power series representation

f (z) = z +1Pj=2

ajzj: (5)

A function f(z) that is analytic (holomorphic) in E, is said to be univalent in E, if it

assumes atmost one value in E. Such a function is also called simple or Schlicht in E:

Denition 2.1.3 A function f (z) is said to be univalent in E, if it provides a one-to-one

mapping between the open unit disk E and the image domain f (E) : Thus a single valued

function f (z) is said to be univalent in E if for z1; z2 2 E;

f (z1) 6= f (z2) implies that z1 6= z2:

The Möbius transformation

L (z) =az + b

cz + d; where a; b; c; d 2 C;

such that ad � bc 6= 0; z 2 E is the simplest example of univalent function. The term

locally univalent is also often used in the literature. We dene it as below.

Denition 2.1.4 A function f (z) is said to be locally univalent or conformal at a point

z0 2 E; if it is univalent in some neighborhood of z0: For analytic function f (z) the

condition f0(z) 6= 0 is equivalent to the local univalence at z0:

Denition 2.1.5 The class S consists of the normalized analytic functions univalent in

E: That is

S = ff : f 2 A, f (z) is univalent in Eg:

The most common example of functions from class S is Koebe function

k (z) =z

(1� z)2= z +

1Pj=2

jzj; z 2 E: (6)

The class S is not closed under addition. However, the class S is preserved under some

11

of the following elementary transformations.

Theorem 2.1.1 If f(z) 2 S; then each of the following function h(z) 2 S:

(i) Conjugation

h(z) = e�i�f(ei�z); � 2 R:

(ii) Rotation

h(z) = f (z):

(iii) Dilation

h(z) =1

tf (tz) ; t 2 (0; 1) :

(iv) Disk automorphism

h(z) =f�z+�

1+�z

�� f (z)�

1� j�j2 f 0 (�)� ; for j�j < 1:

(v) k-fold symmetry

h(z) =�f(zk)

� 1k for k = 1; 2; :::

(vi) Omitted-value transformation

h(z) =f(z)

1� f(z)w

; where w 6= f(z):

Here is the Bieberbachs theorem which states as follows.

Lemma 2.1.1 [10] Let

f : f (z) = z +1Pj=2

ajzj 2 S:

Then ja2j � 2 and this inequality is sharp. Equality holds for some rotation of Koebe

function given by (6).

12

In the followings, we give distortion bounds, for functions in the class S. For reference,

see [11, 15].

Lemma 2.1.2 Let f(z) 2 S and let z = �ei� 2 E; then

�

(1 + �)2� jf(z)j � �

(1� �)2;

and1� �(1 + �)3

��f 0 (z)�� 1 + �

(1� �)3:

These results are sharp. Koebe function and some of its rotations provide equality in the

above relations.

As an application of Bieberbachs theorem, we have the well-known covering theorem due

to Koebe which states that each function f(z) 2 S is an open mapping with f(0) = 0, so

its range contains some disk centered at origin. As early as 1907, Koebe [15] discovered

that the range of all functions in S contains an open disk jwj < � , where � is a positive

constant. The Koebe function shows that � � 4 and Bieberbach [15] later established

Koebes conjecture that � can be taken as 14.

Theorem 2.1.2 [11, 15] Let f(z) 2 S with f(z) 6= w 2 E; then jwj � 14and equality

holds for Koebe function dened by (6).

Di¤erential Subordination

The idea of di¤erential subordination initiated by Lindelöf [25] and was further enriched

by Littlewood [26, 27] and Rogosinski [76, 77] who established the basic results involving

subordination. In very simple terms, a di¤erential subordination in a complex plane is

the generalization of di¤erential inequality on the real line.

Denition 2.1.6 [39] Let s(z) and t(z) be analytic in E. Then s(z) is subordinate to

t(z), written as s(z) � t(z) if there exists a Schwarz function w(z); which is analytic in

E with w(0) = 0 and jw(z)j < 1; such that s(z) = t(w(z)): In particular, when t(z) is

13

univalent, then s(0) = t(0) and s (E) � t (E) :

Convolution

The concept of Convolution or Hadamard product is of fundamental importance which

has many applications in the eld of Geometric Function Theory. It started in 1958 with

the celebrated conjecture regarding the convolution of two convex functions by Polya

and Schoenberg [70]. In the begining, this type of product was named after Jacques

Hadamard who published rst in depth the analysis of this product.

Denition 2.1.7 [82] Let s(z) and t(z) be two analytic functions dened in open unit

disk E, given by

s(z) =1Pj=2

sjzj and t(z) =

1Pj=2

tjzj:

Then convolution (Hadamard product), denoted by (�) of s(z) and t(z) is dened as

H (z) = (s � t) (z) =1Pj=2

sjtjzj; z 2 E:

Convolution has the algebraic properties of ordinary multiplications such as commuta-

tivity and associativity etc. The geometric series

I (z) =1Xj=1

zj =z

1� z ; (7)

acts as the identity element under convolution: (f(z) � I(z)) = f(z) = (I(z) � f(z)) for

all f(z) 2 A:

Theorem 2.1.3 [84] (i) If s(z); t(z) 2 C; then H(z) = s(z) � t(z) 2 C:

(ii) If s(z) 2 S�; t(z) 2 C; then H(z) 2 S�:

(iii) If s(z) 2 K; t(z) 2 C; then H(z) 2 K:

(iv) If s(z) 2 C�; t(z) 2 C; then H(z) 2 C�:

14

Denition 2.1.8 [39] The univalent function q(z) is said to be a dominant of the solutions

of the di¤erential subordination, if r(z) � q(z) for all solutions r(z) satisfying the given

di¤erential subordination. A dominant ~q(z) that satises q(z) � ~q(z) for all dominants

q(z) of the di¤erential subordination is known as the best dominant. It is noted that the

best dominant is unique upto the rotation of the open unit disk E:

Denition 2.1.9 [39] Let a; b and c be complex numbers with c 6= 0;�1;�2; : : : The

function

F (a; b; c; z) = 2F1 (a; b; c; z) =1Xk=0

(a)k (b)k(c)k

zk

k!:

2F1 (a; b; c; z) = 1 +ab

c

z

1!+a (a+ 1) b (b+ 1)

c (c+ 1)

z2

2!+ : : : :; (8)

is called Gaussian hypergeometric function which is analytic in E and satises hyperge-

ometric di¤erential equation

z (1� z)w00 (z) + [c� (a+ b+ 1) z]w0 (z)� abw (z) = 0:

Some of the basic properties of hypergeometric function are as:

2F1 (a; b; c; z) = 2F1 (b; a; c; z) ;

2F1 (a; b; c; z) = (1� z)�a 2F1�a; c� b; c; z

z � 1

�;

2F1 (a; b; b; z) = (1� z)�a : (9)

If Re c > Re b > 0; then

2F1 (a; b; c; z) =� (c)

� (b) � (c� b)

1Z0

tb�1 (1� tz)�a (1� t)c�b�1 dt: (10)

15

2.2 The Caratheodory Class of Functions with Positive Real Part and Some

Related Classes

The class of functions with positive real part plays a crucial role in the Geometric Function

Theory. Its signicance can be seen from the fact that all the simple subclasses of the class

of univalent functions have been dened by using the concept of the class of functions

with positive real part. In this section, we dene the class of functions with positive real

part and we present here some of its interesting properties and also discuss some of its

generalizations.

Denition 2.2.1 [11, 15, 16] The analytic functions p(z) which satisfy the conditions

p (0) = 1 and Re fp (z)g > 0; z 2 E are said to form the class P: That is

(p 2 P : p (z) = 1 +

1Pj=1

pjzj; if and only if Re fp (z)g > 0; z 2 E

):

A function p(z) in P is called a function with positive real part. The Möbius function

Mo (z) =1 + z

1� z = 1 +1Pj=1

2zj; z 2 E; (11)

plays the part of extremal function for this class in many cases. A function p(z) 2 P

need not to be univalent.

Denition 2.2.2 [11, 15] The set P is convex. This mean that if �1 and �2 are non-

negative with �1 + �2 = 1 and p1(z); p2(z) are in P; then

p (z) = �1p1 (z) + �2p2 (z) ;

is also in P:

16

Noshiro Warschawski Theorem

In 1935, Noshiro [64] and Warschawski [95] independently proved a theorem, which is

called Noshiro-Warschawski Theorem and shows a beautiful relationship between the

class P and class S.

Theorem 2.2.1 [64, 95] Suppose that for some real �, Re�ei�f 0 (z)

� 0 for all z in a

convex domain D: Then f (z) is univalent in E:

In the theorems below, we will study the coe¢ cient bounds, growth and distortion results

for functions in the class P . For these results, we refer to [11, 15].

Lemma 2.2.1 Let p (z) 2 P is of the form

p (z) = 1 +1Pn=1

pjzj; (12)

then

jpjj � 2; j = 1; 2; : : : :

Equality holds for some rotation of the Möbius function given by (11).

Lemma 2.2.2 Let p(z) 2 P: Then for jzj = r < 1

1� r1 + r

� Re p (z) � jp (z)j � 1 + r1� r ;

jzp0 (z)j � 2rRe p (z)1� r2 :

These bounds are sharp and equalities hold if and only if p (z) is a suitable rotation of

the Möbius function dened by (11).

Denition 2.2.3 Let P (�) be the class containing the functions p (z) analytic in unit

disk E with Re fp (z)g > � where; 0 � � < 1: If p(z) 2 P (�), then we can write p (z) as

p (z) = (1� �) p1 (z) + �; p1 2 P; z 2 E:

17

Note that P (�) is also a convex set.

Lemma 2.2.3 Let p(z) 2 P (�) and be of the form (12). Then

jpnj � 2 (1� �) ; for all n � 1:

Lemma 2.2.4 Let p(z) 2 P (�) : Then for jzj = r < 1;

1� (1� 2�) r1 + r

� jp (z)j � 1 + (1� 2�) r1� r ; z = re

i�:

Denition 2.2.4 [68] An analytic function p (z) is said to be in the class Pm, if p(0) = 1

and

p(z) =1

2�

2�Z0

1 + ze�it

1� ze�itd�(�), z 2 E;

where �(�) is a non-decreasing real valued function with bounded variation on [0; 2�]

such that for m � 2;2�Z0

d�(�) = 2 and

2�Z0

jd�(�)j � m: (13)

Equivalently for p1 (z) ; p2 (z) 2 P and z 2 E then p (z) 2 Pm if

p (z) =

�m

4+1

2

�p1 (z)�

�m

4� 12

�p2 (z) : (14)

Note that for m = 2; we obtain the class P of functions with positive real part. Also it

is known that Pm is a convex set.

Lemma 2.2.5 If p (z) 2 Pm for z = rei� and m � 2; then we have

1�mr + r21� r2 � Re p(z) �

1 +mr + r2

1� r2 :

These bounds are sharp.

18

Lemma 2.2.6 Let p (z) 2 Pm then we have

jzp0 (z)j � r (m+ 4r +mr2) Re p (z)

(1� r2) (1 +mr + r2) :

For the above two results, see [68].

Theorem 2.2.2 [67] Let Pm(�) be the class of functions p (z) dened by (12) satisfying

the conditions p(0) = 1; 0 � � < 1 and

p(z) =1

2�

2�Z0

1 + (1� 2�) ze�it1� ze�it d�(�), z 2 E:

where �(�) is a real valued function of bounded variation on [0; 2�] and satises the

conditions given by (13). It is known [53] that Pm(�) forms a convex set.

Denition 2.2.5 Let A and B be arbitrary xed numbers with �1 � B < A � 1: A

function p (z) analytic in E with p (0) = 1; belongs to the class P [A;B] if

p (z) � 1 + Az1 +Bz

; z 2 E: (15)

In particular P [A;B] � P [1;�1] =P: Geometrically, a function p (z) 2 P [A;B] if and

only if p (0) = 1 and p (E) lies inside an open unit disk with center 1�AB1�B2 on the real axis

having radius A�B1�B2 with diameter end points p (�1) =

1�A1�B and p (1) =

1+A1+B

for B 6= �1:

The class P [A;B] is connected with the class P of functions with positive real parts by

the relation

p (z) 2 P , (A+ 1) p (z)� (A� 1)(B + 1) p (z)� (B � 1) 2 P [A;B] :

It is known that P [A;B] is a convex set [52]. The class P [A;B] was introduced in [17].

Lemma 2.2.7 Let p (z) 2 P [A;B] ; �1 � B < A � 1: Then for jzj = r < 1; we have

1� Ar1�Br � Re p (z) � jp (z)j �

1 + Ar

1 +Br:

19

Lemma 2.2.8 Let p (z) 2 P [A;B] ; and of the form (12) then

jpnj � A�B:

For the above two results, see [17].

Denition 2.2.6 [55] For �1 � B < A � 1; m � 2; let Pm [A;B] be the class of

functions p (z) analytic in E satisfying p(0) = 1 and

p(z) =1

2�

2�Z0

1 + Aze�it

1 +Bze�itd�(�), z 2 E;

where �(�) is a measure on [0; 2�] and satisfying the conditions given by (13). When

m = 2; we obtain P2 [A;B] = P [A;B] which was introduced by [17]. A function p (z) 2

Pm [A;B] can also be represented as

p (z) =

�m

4+1

2

�p1 (z)�

�m

4� 12

�p2 (z) ;

where p1 (z) ; p2 (z) 2 P [A;B], z 2 E; �1 � B < A � 1 and m � 2:

2.3 Certain Subclasses of Univalent Functions

In this section we shall discuss some subclasses of normalized univalent functions which

are dened by natural geometric conditions. We shall also discuss some interesting prop-

erties including their relationship with the class P of functions with positive real part.

2.3.1 Convex and Starlike Univalent Functions

We can categorize analytic functions into various subclasses on the basis of geometry

of their image domain. These subclasses include the classes of convex, starlike, close-

to-convex and quasi-convex etc. In this section we will study their relationship with

20

the class P: We will investigate connections between geometry of image domain and

analytic properties of these functions along with their mutual relationship, see [11, 15].

The natural generalization of convex functions was provided by Goodman in 1991, by

introducing the concept of uniformly convex and starlike functions. Goodman dened

these classes in the following way by their geometrical mapping properties.

Denition 2.3.1 [11, 15] A domain D � C is called convex if for every pair of points

in the interior of D; the line segment joining them is also in the interior of D: A function

f (z) 2 S is said to be convex in E if the image of E under f (z), that is f (E) is a convex

domain. We can say that any line segment joining any two points in f (E) lies entirely

in f (E) : The class of all convex univalent functions is denoted by C:

Theorem 2.3.1 [90] Let f (z) 2 S: Then f (z) maps E onto a convex domain if and only

if

Re(zf 0 (z))0

f 0 (z)> 0; z 2 E: (16)

or(zf 0 (z))0

f 0 (z)2 P:

The function f (z) = z1�z ; z 2 E acts as extremal function for this class. The function

f (z) = 1+z1�z is a convex function, since it maps E onto a half-plane. The condition for

convexity described in (16) was rst stated and studied in [90].

Denition 2.3.2 [11, 15] A domain D � C is called starlike with respect to a point

w0 = 0; if the line segment joining w0 = 0 to any other point of D lies wholly in D: A

function f (z) 2 S is said to be starlike in E if the image of E under f (z) consists of the

point w0 = 0 is a starlike domain with respect to the origin w0 = 0: That is, any line

segment joining w0 = 0 to every other point in f (E) ; lies entirely in f (E) : The class

of all starlike functions with respect to the origin in E is denoted by S�.

Theorem 2.3.1 [44] Let f (z) 2 S: Then f (z) maps E onto a starlike domain if and

only if

Rezf 0 (z)

f (z)> 0; z 2 E: (17)

21

orzf 0 (z)

f (z)2 P:

For example Koebe function k (z) = z(1�z)2 is a starlike function which plays the role of

extremal function for this class. It is noted that

C � S� � S:

In 1915, Alexander [2] observed the connection between starlike and convex functions as

follows.

f (z) 2 C () zf 0 (z) 2 S�:

Geometric Interpretation of Starlike and Convex Functions

Let �z be a smooth curve with parametrization z (t) = x (t) + iy (t), a � t � b where

x (t) and y (t) are real functions, and z0(t) = x

0(t) + iy

0(t) 6= 0 for t 2 [a; b] : The arc �z

is a directed arc, the direction being determined as t increases. Let �w be the image of

�z under a function f (z) that is analytic on �z. Assume that wo is not on �w: The arc

�w is said to be starlike with respect to wo if arg (w � wo) is a nondecreasing function of

t, that is, ifd

dtarg (w � wo) � 0; t 2 [a; b] : (18)

To convert this inequality to a more useful form we have,

d

dtarg (w � wo) =

d

dtIm ln (w � wo) = Im

d

dtln (w � wo) ;

= Im

�d

dzln (w � wo)

dz

dt

�= Im

�f 0 (z)

f (z)� wodz

dt

�:

22

The image of �z under f (z) is starlike with respect to wo if and only if

Im

�f 0 (z)

f (z)� woz0(t)

�� 0; t 2 [a; b] : (19)

Similarly arc �w is said to be convex if the argument of the tangent to �w is a nonde-

creasing function of t. The direction of the tangent to �z is arg z0(t) and the mapping

rotates this tangent vector through an angle arg f 0 (z) : Thus the arc �w is convex arc if

and only ifd�

dt=

d

dt

harg(z

0(t) f 0 (z))

i� 0; t 2 [a; b] :

The same technique used on equation (18) gives

d

dt

harg(z

0(t) f 0 (z))

i=

d

dtImhln z

0(t) + ln f 0 (z)

i;

= Im

�z00(t)

z0 (t)+

d

dz(ln f 0 (z))

dz

dt

�;

= Im

�z00(t)

z0 (t)+f00(z)

f 0 (z)z0(t)

�:

Suppose that f 0 (z) 6= 0 on �z: Then the image of �z under f (z) is a convex arc if and

only if

Im

�z00(t)

z0 (t)+f00(z)

f 0 (z)z0(t)

�� 0; t 2 [a; b] : (20)

We now specialize these formulas by selecting �z to be the circle Cr : jzj = r: Thus

with the usual orientation, z = reit; 0 � t � 2�: In this case z0 (t) = ireit = iz and

z" (t) = �reit = �z: The inequality (19) becomes

Im

�izf 0 (z)

f (z)� wo

�= Re

�zf 0 (z)

f (z)� wo

�� 0;

and the inequality (20) becomes

Im

�i+ i

zf00(z)

f 0 (z)

�= Re

�1 +

zf00(z)

f 0 (z)

�� 0:

23

The distortion and growth theorems for the class of starlike functions and the class of

convex functions are given as follows and we refer to [11, 15] for these results.

Lemma 2.3.2 Let f (z) 2 S� and is given by (5) : Then for z 2 E;

jajj � j; j � 2:

This result is sharp and equality holds for a suitable rotation of the Koebe function

given by (6) :

Lemma 2.3.3 Let f (z) 2 S�: Then for each jzj = r < 1;

r

(1 + r)2� jf (z)j � r

(1� r)2;

and1� r(1 + r)3

� jf 0 (z)j � 1 + r(1� r)3

:

These inequalities are sharp and equality occurs for some rotation of the Koebe function

given by (6) :

Lemma 2.3.4 Let f (z) 2 C and is given by (5) : Then for z 2 E;

jajj � 1; for all j � 2;

This bound is sharp and equality holds for the function given by (7).

Lemma 2.3.5 Let f (z) 2 C: Then for each jzj = r < 1;

r

1 + r� jf (z)j � r

1� r ;

and1

(1 + r)2� jf 0 (z)j � 1

(1� r)2:

These results are sharp and equality occurs for some rotation of the function given by

(6):

24

Denition 2.3.3 [42] Let f (z) 2 A and z�1f (z) f 0 (z) 6= 0: Then f 2M (�) for � 2 R;

if and only if

Re

�(1� �) zf

0 (z)

f (z)+ �

(zf 0 (z))0

f 0 (z)

�> 0; for all z 2 E:

The above class is also meaningful if we take � to be a complex number, but we assume

� to be real. We see that

M (0) � S� and M (1) = C:

Miller, Mocanu and Read [41] proved that M (�) � S� for � � 0 and M (�) � C for

� � 1:

Denition 2.3.4 [75] Let f (z) 2 A be analytic in E and is given by (5), then f (z) is

said to be convex of order � in E if for all z 2 E; 0 � � < 1;

(zf 0 (z))0

f 0 (z)2 P (�) ; z 2 E:

The class of such functions is denoted by C (�). It is clear that C (�) � C and C (0) = C:

Marx [36] and Strohhacker [89] proved that if f (z) 2 C (0) ; then f 2 S��12

�:

Denition 2.3.5 [75] Let f (z) 2 A be analytic in E and is given by (5). Then f (z)

is said to be starlike of order � in E, if for all z 2 E; 0 � � < 1;

zf 0 (z)

f (z)2 P (�) ; z 2 E:

The class of such functions is denoted by S� (�). It is clear that S� (0) = S�:We observe

that f (z) 2 C (�) if and only if zf 0 (z) 2 S� (�) :

Denition 2.3.6 [96] A function f (z) 2 A is called convex function of complex order �

if and only if �1 +

1

�

�(zf 0 (z))0

f 0 (z)

��2 P; for � 2 Cnf0g:

We denote such a class of functions as C�: For � = 1� �; 0 � � < 1; C� = C (�) ; class

25

of convex functions of order �:

Denition 2.3.7 [43] A function f (z) 2 A is called starlike function of complex order

� if and only if �1 +

1

�

�zf 0 (z)

f (z)� 1��

2 P; for � 2 Cnf0g:

We denote such a class of functions as S��. For � = 1� �; 0 � � < 1; S�� = S� (�) ; class

of starlike functions of order �:

Denition 2.3.8 [17] Let �1 � B < A � 1, then a function f (z) 2 A is said to belong

to the class C [A;B], if and only if

(zf 0 (z))0

f 0 (z)2 P [A;B] :

Denition 2.3.9 [17] Let �1 � B < A � 1, then a function f (z) 2 A is said to belongs

to the class S� [A;B], if and only if

zf 0 (z)

f (z)2 P [A;B] :

It is noted that Alexanders relation holds for these classes, that is,

f (z) 2 C [A;B] () zf 0 (z) 2 S� [A;B] :

Special values given to A and B lead to some well-known classes of analytic functions,

just as

C [1;�1] = C � S;

S� [1;�1] = S� � S:

26

The class K of Close-to-Convex Functions

We now focus our attention to an interesting subclass of univalent functions introduced

by Kaplan [18] in 1952 which contains S� and has a simple geometric characterization.

This class of close-to-convex functions is denoted by K:

Denition 2.3.10 [18] Let f (z) 2 A with f 0 (z) 6= 0: Then f (z) 2 K, if there is a

convex function h (z) such that

Ref 0 (z)

h0 (z)> 0; z 2 E:

Since H (z) = zh0 (z) 2 S� when h (z) 2 C; so we have

Rezf 0 (z)

H (z)> 0; z 2 E:

Observe that starlike functions are close-to-convex. Thus we can summarized the chain

of proper inclusions

C � S� � K � S:

It was proved in [18] that every close-to-convex function is univalent.

Geometric Interpretation

Let f (z) 2 A and Cr be the image of the circle jzj = r; where 0 < r < 1 under f (z) :

Then f (z) 2 K, if and only if , as � increases, either in the counterclockwise direction

or clockwise direction the tangent direction arg�@@�f 0�rei��along the Cr must not

decrease by as much as � from any previous value.

The following theorem, due to Kaplan [18] gives the characterization for f (z) to be in

27

the class K:

Theorem 2.3.2 Let f (z) 2 A with f 0 (z) 6= 0 in E . Then f (z) 2 K if and only if

2�Z0

Re

�(zf 0 (z))0

f 0 (z)

�d� > ��; z = rei�; (21)

for each r (0 < r < 1) and 0 � �1 < �2 � 2�: The condition f 0 (z) 6= 0 in E must be

included in this theorem, because the function f (z) = zm; (m > 1) satises the inequality

(21) but is not univalent and hence not close-to- convex.

Figure 2.1: Domain of close-to-convex functions

In Figure 2.1 change of argument of tangent lines is slightly greater than ��. Thus such

a hairpin turnis permitted, provided it does not make a complete reversal direction.

The Class C* of Quasi-Convex Functions

Analogous to the class K of close-to-convex functions, Noor [49] introduced the class C�

of quasi-convex functions.

28

Denition 2.3.11 [60] Let f (z) 2 A with f 0 (z) 6= 0 in E: Then f (z) 2 C�; if for some

g (z) 2 C

Re(zf 0 (z))0

g0 (z)> 0; z 2 E:

We note that when g (z) = f (z) then f (z) 2 C: That is C � C�: It was proved in [60]

that every f (z) 2 C� is close to convex and hence univalent. Thus

C � C� � K � S:

Further it was proved that C� has no inclusion relationship with S�: An Alexander type

relation also holds between the classes K and C�. That is, f (z) 2 C�; if and only if,

zf 0 (z) 2 K:

Close-to-Convex and Quasi-Convex Functions of Order Type �

Libera [24] introduced the terminology of order and type together for the class of close-

to-convex functions. Here we dene it as follows.

Denition 2.3.12 [24] A function f (z) 2 A is said to be close-to-convex of order � type

�; where 0 � � < 1 and 0 � � < 1 if and only if there exists a function g (z) 2 S� (�)

such thatzf 0 (z)

g (z)2 P (�) ; z 2 E:

We denote this class as K (�; �) : It is clear that K (�; 0) = K (�) and K (0; 0) = K:

Since K (�; �) � K (0; 0) ; we see that K (�; �) is a subclass of S; hence univalent.

In 1987, Noor [54] introduced the class of quasi-convex of order � type � which we dene

as follows.

Denition 2.3.13 [54] A function f (z) 2 A is said to be quasi-convex of order � type

�; where 0 � � < 1 and 0 � � < 1; if and only if, there exists a function g (z) 2 C (�)

29

such that(zf 0 (z))0

g0 (z)2 P (�) ; z 2 E:

We denote this class as C� (�; �) : Clearly f (z) 2 C� (�; �) ; if and only if zf 0 (z) 2

K (�; �) : Also C� (0; 0) = C�:

2.4 Functions with Bounded Radius and Bounded Boundary Rotations

This section introduces some subclasses of analytic functions such as functions of bounded

boundary and bounded radius rotations. Also some of their generalized classes will be

discussed here.

Functions with bounded boundary rotations

Denition 2.4.1 [32, 66] For a simple closed domain with smooth boundary, the bound-

ary rotation is dened as the total variation of the argument of the boundary tangent

vector (whenever such a tangent vector exists) which can be greater or equal to 2�: A

functions f(z) analytic and locally univalent in E is said to be of bounded boundary

rotation if its range has bounded boundary rotation. Let Vm; m � 2 denotes the class of

analytic functions f(z) dened in E, given by (5) and which maps E conformally onto

an image domain of bounded boundary rotation at most m�. An analytic representation

for functions f(z) in the class Vm is given by

2�Z0

��Re�(zf 0 (z))0f 0 (z)�� d� � m�, m � 2.

Brannan [6] gave another representation for the functions of the class Vm in the terms of

starlike functions as follows. An analytic functions f(z) 2 Vm if and only if there exist

30

s1 (z) ; s2 (z) 2 S� such that

zf 0 (z) =(s1 (z))

(m+24 )

(s2 (z))(m�24 )

; m � 2; z 2 E: (22)

Equivalently, it can be written as f(z) 2 Vm if and only if

(zf 0 (z))0

f 0 (z)2 Pm; m � 2; z 2 E:

Note that V2 � C, the class of convex functions. It was proved by Paatero [66] that,

for 2 � m � 4, Vm � S and Renyi [74] proved that jajj � j for V4: Later, Pinchuk [68]

and Brannan [6] found that V4 is properly contained in the class K of close-to-convex

functions. However, for m > 4, Vm consists of non-univalent functions. It was proved by

Kirwan [22] that the radius of univalence of Vm for m > 4 is tan �k . For arbitrary m � 2,

Lowner [32] obtained the sharp distortion result

(1� r)m2�1

(1 + r)m2+1� jf 0(z)j � (1 + r)

m2�1

(1� r)m2+1, jzj = r < 1,

for all f(z) 2 Vm. This result is sharp and equalities occurs for certain rotations of the

the wedge mapping

Fm(z) =1

m

"�1 + z

1� z

�m2

� 1#;

which plays the part of the Koebe function for the class Vm:

Functions with bounded radius rotations

Denition 2.4.2 For a simple closed domain with smooth boundary, the radius rotation

is dened as the total variation of the argument of the radial vector (whenever such a

radial vector exists) which can be greater or equal to 2�: A functions f(z) analytic and

locally univalent in E is said to be of bounded radius rotation if its range has bounded

31

radius rotation. Let Rm; m � 2 denotes the class of analytic functions f(z) dened in

E and given by (12) and which maps E conformally onto an image domain of bounded

radius rotation at most m�. For the xed m � 2 an analytic representation for functions

f(z) in the class Rm is given by

2�Z0

��Re zf 0(z)f(z)�� d� � m�, m � 2:

Equivalently, it can be written as f(z) 2 Rm if and only if

zf 0(z)

f(z)2 Pm; z 2 E:

The class Rm was introduced by Tammi [91] in 1952. We note that R2 � S�, the class

of starlike functions with respect to origin. Also Vm � Rm; since the radius rotation of a

function is never greater than its boundary rotation. It is clear that

f(z) 2 Vm , zf 0(z) 2 Rm: (23)

The classes Vm [A;B] and Rm [A;B] were introduced by Noor [51]. These classes are

related to the class Pm [A;B] :

Denition 2.4.3 Let f (z) 2 A dened by (12). Then for �1 � B < A � 1 and m � 2;

f (z) 2 Vm [A;B] if and only if

(zf 0 (z))0

f 0 (z)2 Pm [A;B] ; z 2 E:

Denition 2.4.4 Let f (z) 2 A dened by (12). Then for �1 � B < A � 1 and m � 2;

f (z) 2 Rm [A;B] if and only if

zf 0(z)

f(z)2 Pm [A;B] ; z 2 E:

32

It is observed that V2 [A;B] � C [A;B] and R2 [A;B] � S� [A;B] : Taking A = 1 and

B = �1; we have the classes of Vm and Rm already discussed in section 2.4.

Starlikeness and convexity are hereditary properties in the sense that every starlike (con-

vex) function maps each disk jzj < r < 1 onto starlike (convex) domain. However Brown

[8] showed that it is not always true that f (z) 2 S� maps each disk jz � z0j < � < 1� jz0j

onto a domain starlike with respect to f (z0) : He proved that the result is true for each

f (z) 2 S and for su¢ ciently small disk in E: This motivates the denition of uniformly

starlike function, though it was introduced independently of the work of Brown [8].

2.5 Uniformly Convex, Uniformly Starlike and related Functions

Denition 2.5.1 [13, 14] A function f (z) 2 S is uniformly convex ( starlike) if for every

circular arc � contained in E with center z0 2 E the image arc f (�) is convex ( starlike).

The classes of uniformly convex and uniformly starlike functions are denoted by UCV

and UST respectively.

The following two-variable analytic characterization of the class UCV is important for

obtaining information about functions in this class.

Denition 2.5.2 [13] A function f (z) belongs to UCV if and only if

Re

�1 + (z � z0)

f 00(z)

f 0(z)

�� 0; z; z0 2 E: (24)

It is evident that UCV � C: However, by taking z0 = �z in (24) , it is clear that

UCV � C�12

�: Also if we take z0 = 0; we obtain well known class C of convex functions.

Denition 2.5.3 [14] A function f (z) belongs to UST if and only if

Re

�(z � z0) f 0 (z)f (z)� f (z0)

�� 0; z; z0 2 E: (25)

By taking z0 = �z in (25), we obtain the class of starlike functions with respect to

symmetric points [86]. Similarly if we take z0 = 0 in (25), we obtain class S� of star-

33

like functions. Unlike the uniformly starlike functions, uniformly convex functions admit

a one-variable characterizaction. The one-variable characterizaction obtained indepen-

dently by Ronning[81] , and Ma and Minda [35] is the following result.

Theorem 2.5.1 Let f (z) 2 A. Then f(z) 2 UCV if and only if

Re

�(zf 0(z))0

f 0 (z)

�>

��(zf 0(z))0f 0 (z) � 1�� ; z 2 E: (26)

A class closely related to the class UCV is the class of parabolic starlike functions dened

below.

Denition 2.5.4 [81] The class Sp of parabolic starlike functions consists of the functions

f (z) 2 A, satisfying

Re

�zf 0(z)

f (z)

�>

��zf 0(z)f (z) � 1�� ; z 2 E: (27)

In other words, the class Sp consists of the functions f (z) = zF0(z) where F 2 UCV: It

turns out (see [13, 80]) that there is no inclusion between UST and Sp: That is

UST * Sp and Sp * UST:


To give a nice geometric interpretation of UCV and Sp, let

= fw 2 C : Rew > jw � 1jg ; (28)

=�w : w = u+ iv and v2 < 2u� 1

:

The set is the interior of of the parabola

(Imw)2 = 2Rew � 1;

34

and is therefore symmetric with respect to real axis and has�12; 0�as a vertex. Conditions

(26) and (27) respectively imply that (zf0(z))0

f 0(z) andzf 0(z)f(z)

lie in the interior of parabolic

region for all values of z: That is, f 2 UCV , (zf0(z))0

f 0(z) 2 and f 2 Sp ,zf 0(z)f(z)

2 .

Conic Domain

In 1999, Kanas and Wisniowska [19, 20] introduced the conic domain k; k � 0. This

domain is dened as follows:

k = fw : Rew > k jw � 1jg ; (29)

=

�w = u+ iv : u > k

q(u� 1)2 + v2

�:

Then the region k is elliptic for k > 1; parabolic for k = 1; and hyperbolic for ,

0 < k < 1: The region 0 is right half plane as shown in Fig. 2.2.

Figure 2.2: Boundaries of conic region dened by k

The functions which play the role of extremal functions for these conic regions are given

35

as:

pk(z) =

8>>>>>>>>>>>>>>>>>:

1+z1�z , k = 0,

1 + 2�2

�log 1+

pz

1�pz

�2, k = 1,

1 + 21�k2 sinh

2��

2�arccos k

�arctanh

pz�, 0 < k < 1,

1 + 1k2�1 sin

0@ �2R(t)

u(z)ptR

0

1p1�x2

p1�(tx)2

dx

1A+ 1k2�1 , k > 1.

(30)

where u(z) = z�pt

1�ptz, t 2 (0; 1), z 2 E and z is chosen such that k = cosh

��R0(t)4R(t)

�, R(t)

is the Legendres complete elliptic integral of the rst kind and R0(t) is complementary

integral of R(t):

Denition 2.5.5 [19] Let k � 0. A function f (z) 2 S is called k�uniformly convex

in E if the image of every circular arc � contained in the unit disk E, with center z0,

jz0j � k; is convex. For any xed k � 0, the class of all k�uniformly convex functions is

denoted by k � UCV:

Theorem 2.5.2 [19] Let f (z) 2 A; and 0 � k k

��(zf 0(z))0f 0 (z) � 1�� ; k � 0; z 2 E:

Interestingly, the class of k�uniformly convex functions unies the class of convex func-

tions (k = 0) and the class of uniformly convex functions (k = 1) : The class k � ST is

dened as follows.

Denition 2.5.6 [20] Let f (z) 2 A: Then f (z) is in the class k � ST , if and only if

Re

�zf 0(z)

f (z)

�> k

��zf 0(z)f (z) � 1�� ; k � 0; z 2 E:

It is important to note that for 0 � ST � S and 1 � ST � ST: The Alexander relation

holds for these classes. In fact, the class k � ST was dened from k � UCV by mean of

Alexander relation as

k � ST = fh : h (z) = zf 0 (z) ; f (z) 2 k � UCV g :

36


Geometrically, a function f (z) 2 A is said to be in the class k � UCV (or k � ST ) ; if

and only if the function (zf0(z))0

f 0(z)

�or zf

0(z)f(z)

�takes all values in the conic domain k: In

other words we can say f (z) 2 k � UCV and k � ST respectively if and only if

(zf 0(z))0

f 0 (z)� pk(z); z 2 E; k � 0;

andzf 0(z)

f (z)� pk(z); z 2 E; k � 0:

A function f (z) 2 A and of the form (5) is in the class k � UCV; if it satises the

condition1Xn=2

n fn+ k (n� 1)g janj < 1; k � 0: (31)

A function f (z) 2 A and of the form (5) is in the class k�ST; if it satises the condition

1Xn=2

fn+ k (n� 1)g janj < 1; k � 0: (32)

For these inequalities, we refer to [19].

2.6 Bazileviµc Functions

Denition 2.6.1 If g (z) is starlike (with respect to origin) in E; h (z) is analytic in E

with Reh (z) > 0; � is any real number and � > 0 then

f (z) =

8

has been proved by Bazileviµc [3] to be analytic and univalent in E: The powers appearing

in the formula are meant as principal values. We denote the class of functions dened by

(33) by B0(�; �; g; h) : If we put � = 0 in (33) then we have

f (z) =

8 0; z 2 E:

This class was introduced by Obradovic. Uptil now, this class was studied in a direction

of nding necessary condition over � that embeds this class into the class of univalent

functions or its subclasses. This still remains an open problem.

2.7 Certain linear operators dened in terms of convolution

The study of operators can be traced back to 1916 provided by Alexander. Later, Libera

38

(1965) discussed another integral operator and studied its e¤ects on various classes of

univalent functions. Bernardi generalized this operator and investigated its interesting

aspects. A. E. Livingston observed the converse case of Liberas operator. It can be

seen easily that such operators can be interpreted in terms of convolution. The study

of operators, plays an important role in Geometric Function Theory. A large number of

classes of analytic functions are dened by means of di¤erent operators. In this section,

we present a short survey on some operators which are helpful in our later study.

Bernardi Integral Operator

For a function f (z) 2 A; and given by (5) , we consider the integral operator

F (z) = Ic (f (z)) =c+ 1

zc

zZ0

tc�1f (t) dt; c > �1;

= f (z) �1Xj=2

�c+ 1

c+ j

�zj:

The operator Ic, when c 2 N was introduced by Bernardi [4]. In particular, the operator

I1 was studied earlier by Libera [23] and Livingston [31] .

Ruscheweyh Derivative

Let f (z) 2 A and be given by (5). Denote by D� : A ! A; the operator dened by

D�f (z) =z

(1� z)�+1� f (z) ; for � 2 No = f0; 1; 2; :::g;

= z +

1Xj=2

(� + j � 1)!�! (j � 1)! z

j � f (z) :

It is obvious that D0f (z) = f (z), D1f (z) = zf0(z) and D�f (z) =

z(z��1f(z))(�)

�!for all

39

� 2 No:

The following identity can easily be settled

z (D�f (z))0 = (n+ 1)D�+1f (z)� nD�f (z) :

The operator D� : A ! A was originally introduced by Ruscheweyh [83] and named as

�th-order Ruscheweyh derivative by Al-Amiri [1] .

Noor Integral Operator

Analogous to Ruscheweyh Derivatives of order �, Noor [50], and Noor and Noor [59]

dened and studied an integral operator I� : A ! A; for f (z) 2 A dened by (5) as

follows.

Let f� (z) = z(1�z)�+1 ; (� 2 No) and f�1� (z) be dened such that

f� (z) � f�1� (z) =z

(1� z)2:

Then

I�f (z) = f� (z) � f�1� (z) ="

z

(1� z)�+1

#�1� f (z) :

We note that I0f (z) = zf0(z), I1f (z) = f (z).

The following identity can be easily settled:

z (I�+1f (z))0 = (� + 1) I�f (z)� �I�+1f (z) :

The operator I�f (z)) was introduced by Noor [50] and Liu [28] named it "Noor integral

operator" of f(z) of order �:

40

Carlson and Sha¤er Operator

Let �(a; c; z) be the incomplete beta function dened by

� (a; c; z) = z +

1Xj=1

(a)j(c)j

zj; (z 2 E; c 6= 0;�1;�2; :::) :

where (x)j is the shifted factorial dened in terms of a gamma function � by

(x)j =� (x+ j)

� (x)=

8

Lemma 2.8.1 [39] Let u = u1 + iu2, v = v1 + iv2 and (u; v) be a complex valued

function satisfying the conditions,

(i) (u; v) is continuous in a domain D � C2;

(ii) (1; 0) 2 D and Re (1; 0) > 0;

(iii) Re (iu2; v1) � 0; whenever (iu2; v1) 2 D and v1 � �12 (1 + u22) :

If a function h (z) = 1 + c1z + � � � is analytic in E such that (h(z); zh0(z)) 2 D and

Re (h(z); zh0(z)) > 0 for z 2 E; then Reh(z) > 0 in E:

Lemma 2.8.2 [84] Let f(z) 2 C and g(z) 2 S�: Then for any analytic function F (z)

with F (0) = 1 in E;f � Fgf � g (E) � coF (E) ;

where coF (E) denotes the closed convex hull of F (E) (the smallest convex set which

contains F (E)).

Lemma 2.8.3 [40] If �1 � B < A � 1; � > 0 and the complex number satises

Re fg � �� (1� A) = (1�B) ; then the di¤erential equation

q (z) +zq0 (z)

�q (z) + =1 + Az

1 +Bz; z 2 E;

has a univalent solution in E given by

q (z) =

8>>>>>>>>>>>>>>>:

z�+(1+Bz)�(A�B)=B

�

zZ0

t�+�1(1+Bt)�(A�B)=Bdt

� �; B 6= 0;

z�+e�Az

�

zZ0

t�+�1e�Atdt

� �; B = 0:

(36)

If h (z) = 1 + c1z + c2z2 + : : : is analytic in E and satises

h (z) +zh0 (z)

�h (z) + � 1 + Az1 +Bz

; z 2 E;

42

then

h (z) � q (z) � 1 + Az1 +Bz

;

and q (z) is the best dominant.

Lemma 2.8.4 [97] Let " be a positive measure on [0; 1] : Let g(z) be a complex-valued

function dened on E � [0; 1] such that g (:; t) is analytic in E for each t 2 [0; 1] and

g (z; :) is "-integrable on [0; 1] for all z 2 E: In addition, suppose that Re g (z; t) >

0; g (�r; t) is real and Re f1=g (z; t)g � 1=g (�r; t) for jzj � r < 1 and t 2 [0; 1] : If

g (z) =

1Z0

g (z; t) d"t; then Re f1=g (z)g � 1=g (�r) :

Lemma 2.8.5 [98] Let a; b and c 6= 0;�1;�2 : : : be complex numbers. Then, for Re c >

Re b > 0;

(i) 2F1 (a; b; c; z) =� (c)

� (c� b) � (b)

1Z0

tb�1 (1� t)c�b�1 (1� tz)�a dt:

(ii) 2F1 (a; b; c; z) = 2F1 (b; a; c; z) :

(iii) 2F1 (a; b; c; z) = (1� z)�a 2F1�a; c� b; c; z

z � 1

�:

Lemma 2.8.6 [29] Let �1 � B1 � B2 < A2 � A1 � 1: Then

1 + A2z

1 +B2z� 1 + A1z1 +B1z

:

Lemma 2.8.7 [30] Let F (z) be analytic and convex in E. If f(z); g(z) 2 A and

f(z); g(z) � F (z): Then

�f(z) + (1� �) g(z) � F (z); 0 � � � 1:

Lemma 2.8.8 [77] Let f (z) =P1

k=0 akzk be analytic in E and F (z) =

P1k=0 bkz

k be

43

analytic and convex in E. If f(z) � F (z); then

jakj � jb1j ; (k 2 N) :

Lemma 2.8.9 [34] If p (z) = 1 + p1z + p2z2 + : : : is a function with positive real part in

E; then

��p2 � vp21�� 8>>>>>:�4v + 2; v � 0;

2; 0 � v � 1;

4v � 2; v � 1:

When v < 0 or v > 1; equality holds if and only if p (z) is 1+z1�z or one of its rotations.

If 0 < v < 1; then equality holds if and and only if p (z) = 1+z2

1�z2 or one of its rotations.

If v = 0; equality holds if and only if p (z) =�12+ �

2

�1+z1�z +

�12� �

2

�1�z1+z

; (0 � � � 1) or

one of its rotations. If v = 1; equality holds if and only if p(z) is the reciprocal of one

of the functions such that equality holds in the case of v = 0: Although the above upper

bound is sharp when 0 < v < 1; it can improved as follows:

��p2 � vp21��+ v jp1j2 � 2; (0 < v � 1=2) ;and ��p2 � vp21��+ (1� v) jp1j2 � 2; (1=2 < v � 1) :Lemma 2.8.10 [72] If p (z) = 1 + p1z + p2z2 + : : : is a function with positive real part

in E; then for v; a complex number

��p2 � vp21�� 2max (1; j2v � 1j) :This result is sharp for the functions

ho (z) =1 + z2

1� z2 ; h1 (z) =1 + z

1� z :

44

Note that all the references for denitions, theorems and lemmas are given and if there

is any missing it can be seen in [11, 15]. It is also important to note that nothing is

produced by author himself in the second chapter.

45

Chapter 3

Generalized Janowski Functions

46

3.1 Introduction

Let H denote the class of functions w, analytic in the unit disk E = fz : jzj < 1g and of

the form

w (z) =1Pj=1

hjzj = h1z + h2z

2 + ::::;

satisfying the conditions jw(0)j = 0 and jw(z)j < 1 for all z 2 E: Based on the class H,

Janowski [17] dened the class P [A;B] and Pinchuk [68] dened class Pm: Noor combined

them to dened a new class Pm [A;B] : In this chapter, we dene certain new classes of

analytic functions using Janowski functions. New Classes Rm [A;B; �] and Vm [A;B; �]

are also introduced in this chapter by using the various known classes. Some results such

as inclusion result, radius problem, coe¢ cients bounds and integral operators for these

subclasses of analytic functions are investigated. Further we study arc length problem for

the class Pm [A;B; �] : Also it is shown that some of these results are sharp. For di¤erent

choices of parameters we present the relationship of previously known classes with these

classes. The results in this chapter have been published in Journal of World Applied

sciences, see [62].

Denition 3.1.1 [69] For , 0 � � < 1; a function p (z) is said to be in the class P [A;B; �]

if it is analytic in E with p(0) = 1 and

p (z) � 1 + [(1� �)A+ �B] z1 +Bz

; �1 � B < A � 1 and z 2 E,

or equivalently

p (z) 2 P [A;B; �], p (z) = (1� �) po (z) + �; (37)

where po (z) 2 P [A;B] : We observe that P [A;B; 0] = P [A;B] and P [1;�1; 0] = P:

Also P [A;B; �] is a convex set.

47

Remark 3.1.1 As p (z) 2 P [A;B] ; if and only if, for �1 � B < A � 1,

p (z) � 1 + Az1 +Bz

; z 2 E:

By using a more general bilinear transformation h (z) = 1+A1z1+B1z

; with A1 2 C; B1 2 [�1; 0]

and A1 6= B1; we can dene the class P [A1; B1] of analytic function p (z) with

p (z) � 1 + A1z1 +B1z

; z 2 E:

If we take A1 = [(1� �)A+ �B], B1 = B; 0 � � < 1 and �1 � B < A � 1, then

P [A;B; �] = P [A1; B1] :

Let p (z) = 1 +1Pj=1

pjzj 2 P [A;B; �] : Then, for all n � 1;

jpjj � (1� �) (A�B) :

Denition 3.1.2 For 0 � � < 1 and m � 2; p (z) 2 Pm [A;B; �] ; if and only if, there

exists po (z) 2 Pm [A;B], such that

p (z) = (1� �) po (z) + �; z 2 E;

or equivalently let Pm [A;B; �] denote the class of functions p (z) that are analytic in E

and are represented by

p (z) =

�m

4+1

2

�p1 (z)�

�m

4� 12

�p2 (z) ; (38)

where p1 (z) ; p2 (z) 2 P [A;B; �], z 2 E; �1 � B < A � 1, 0 � � < 1 and m � 2: We

note that P2 [A;B; 0] � P [A;B] and Pm [1;�1; 0] is the class Pm rst introduced and

48

studied in [68] .

3.2 Main Results

Theorem 3.2.1 Let p (z) 2 Pm [A;B; �] : Then

1

(1�B2r2)

h1� m

2(1� �) (A�B) r �

�(1� �)AB + �B2

r2i

� Re p (z) � 1(1�B2r2) [1 +

m

2(1� �) (A�B) r

��(1� �)AB + �B2

r2]:

This result is sharp.

Proof. Janowski [17] proved that, for p (z) 2 P [A;B] ;

1� Ar1�Br � Re p (z) � jp (z)j �

1 + Ar

1 +Br: (39)

Using (37), (38) and (39), we have

p (z) =

�m

4+1

2

�p1 (z)�

�m

4� 12

�p2 (z) :

=

�m

4+1

2

�[(1� �)p01 (z) + �]�

�m

4� 12

�[(1� �)p02 (z) + �] :

Therefore

Re p (z) ��m

4+1

2

�[(1� �) Re p01 (z) + �]�

�m

4� 12

�[(1� �) Re p02 (z) + �] ;

which implies

Re p (z) ��m

4+1

2

��(1� �)

�1� Ar1�Br

�+ �

��m

4� 12

��(1� �)

�1 + Ar

1 +Br

�+ �

�;

49

which gives

=

�m4+ 1

2

�(1 +Br) (1� Ar + �Ar � �Br)�

�m4� 1

2

�(1�Br) (1 + Ar � �Ar + �Br)

(1�Br) (1 +Br) ;

=1� m

2(A� �A+ �B �B) r + (AB � �AB + �B2) r2

1�B2r2 ;

=1

(1�B2r2)

h1� m

2(1� �) (A�B) r �

�(1� �)AB + �B2

r2i:

For the upper bound, we note that

Re p (z) ��m

4+1

2

�max

P12P [A;B][(1� �) Re p1 (z) + �]�

�m

4� 12

�min

P22P [A;B][(1� �) Re p2 (z) + �] ;

which gives

Re p (z) ��m

4+1

2

��(1� �)

�1 + Ar

1 +Br

�+ �

��m

4� 12

��(1� �)

�1� Ar1�Br

�+ �

�;

and this implies that

Re p (z) �1 + m

2(A� �A+ �B �B) r � (AB � �AB + �B2) r2

1�B2r2 ;

=1

(1�B2r2)

h1 +

m

2(1� �) (A�B) r �

�(1� �)AB + �B2

r2i:

By taking

pi =1 + Aiz

1 +Biz; i = 1; 2;

in (37), it easily follows that this result is sharp.

Theorem 3.2.2 Let p (z) 2 Pm [A;B; �] and be given by (12) then

jpjj �m (A�B) (1� �)

2; 8n � 1:

50

This result is sharp.

Proof. Since p (z) 2 Pm [A;B; �] ; then from (37) and (38), we have

p (z) =

�m

4+1

2

�[(1� �)p1 (z) + �]�

�m

4� 12

�[(1� �)p2 (z) + �] ;

where p (z) = 1 +1Pj=1

pjzj; p1 (z) = 1 +

1Pj=1

p0jzj 2 P [A;B] and p2 (z) = 1 +

1Pj=1

p00j zj 2

P [A;B] : Thus

1+

1Xj=1

pjzj =

�m

4+1

2

�"(1� �)

(1 +

1Xj=1

p0

jzj

)+ �

#��m

4� 12

�"(1� �)

(1 +

1Xj=1

p00

j zj

)+ �

#;

and so

pj =

�m

4+1

2

�(1� �)p0j �

�m

4� 12

�(1� �)p00j ,

Now using well-known coe¢ cients bounds given in Lemma 2.2.8, we have

jpnj ��m

4+1

2

�(1� �) (A�B) +

�m

4� 12

�(1� �) (A�B) ;

which gives

jpnj �m (A�B) (1� �)

2:

Hence this gives the required result.

Theorem 3.2.3 The class Pm [A;B; �] is a convex set.

Proof. Let p (z) ; q (z) 2 Pm[A;B; �]. Then there exist functions p1 (z) ; p2 (z) ; q1 (z) and

q2 (z) 2 P [A;B; �] such that

p (z) =

�m

4+1

2

�p1 (z)�

�m

4� 12

�p2 (z) ;

and

q (z) =

�m

4+1

2

�q1 (z)�

�m

4� 12

�q2 (z) :

51

Now for 0 � � � 1; consider

(1� �) p (z) + �q (z) = (1� �)��

m

4+1

2

�p1 (z)�

�m

4� 12

�p2 (z)

�+�

��m

4+1

2

�q1 (z)�

�m

4� 12

�q2 (z)

�:

Therefore, we have

(1� �) p (z) + �q (z) =�m

4+1

2

�[(1� �) p1 (z) + �q1 (z)]

��m

4� 12

�[(1� �) p2 (z) + �q2 (z)];

=

�m

4+1

2

�h1 (z)�

�m

4� 12

�h2 (z) :

where h1 (z) = (1� �) p1 (z) + �q1 (z) and h2 (z) = (1� �) p2 (z) + �q2 (z) both belong

to P [A;B; �] being a convex set. Hence Pm [A;B; �] is a convex set.

Theorem 3.2.4 Let p (z) 2 Pm [A;B; �] and is given by (12). Then with z = rei� 2 E;

(i) 12�

2�Z0

��p �rei��2 d� � 4+[fm(1��)(A�B)g2�4]r24(1�r2) :

(ii) 12�

2�Z0

��p0 �rei�� d� � m(1��)(A�B)2(1�B2r2) :

Proof. (i) Using the Parsevals identity, we have

1

2�

2�Z0

��p �rei��2 = 1Xj=0

jpnj2 r2j;

which gives

1

2�

2�Z0

��p �rei��2 � 1 + m2 (1� �)2 (A�B)24

1Xn=1

r2j;

52

where we have used Theorem 3.2.2. Thus we have

1

2�

2�Z0

��p �rei��2 � 4 + [fm (1� �) (A�B)g2 � 4] r24 (1� r2) :

(ii) Since p (z) 2 Pm [A;B; �] so we have

p (z) =

�m

4+1

2

�[(1� �)p1 (z) + �]�

�m

4� 12

�[(1� �)p2 (z) + �] ;

where p1; p2 2 P [A;B] ;

p0(z) =

�m

4+1

2

�h(1� �)p01 (z)

i��m

4� 12

�h(1� �)p02 (z)

i: (40)

Now, for all pi 2 P [A;B] ; we can write

pi (z) =1 + Aw (z)

1 +Bw (z);

which gives

p0

i (z) =(A�B)w0 (z)[1 +Bw (z)]2

; i = 1; 2:

So

1

2�

2�Z0

��p0i �rei�� d� = 12�2�Z0

(A�B)w0�rei��

[1 +Bw (rei�)]2d�; i = 1; 2: (41)

� A�B1�B2r2 :

From (40) and (41), we have

1

2�

2�Z0

��p0 �rei�� d� � �m4+1

2

�(1��) 1

2�

2�Z0

��p01 (z)�� d�+�m4 � 12�(1��) 1

2�

2�Z0

��p02 (z)�� d�

53

��m

4+1

2

�(1� �) A�B

1�B2r2 +�m

4� 12

�(1� �) A�B

1�B2r2 ;

which implies

=m (1� �) (A�B)2 (1�B2r2) :

This completes the proof.

We shall assume throughout, unless stated otherwise, that �1 � B < A � 1; m � 2;

0 � � < 1 and z 2 E.

Denition 3.2.1 [69] A function f (z) 2 A is said to be in the class S� [A;B; �] ; if and

only if,zf 0(z)

f(z)2 P [A;B; �] ; z 2 E:

Note that S� [A;B; 0] � S� [A;B] and S� [1;�1; 0] � S�:

Lemma 3.2.1 [69] If f (z) 2 S� [A;B; �] ; then8

It is clear that

f (z) 2 Vm [A;B; �] if and only if zf0(z) 2 Rm [A;B; �] : (42)

We note that Vm [1;�1; 0] � Vm and Rm [1;�1; 0] � Rm are the well-known classes of

functions with bounded boundary and bounded radius rotation respectively; see section

2.4 of chapter 2 . The class R2 [A;B; �] � S� [A;B; �] has been studied in some details

in [69].

Theorem 3.2.5 Let f (z) be analytic in E with f (0) = 0 and f 0 (0) = 1: Then f (z) 2

Rm [A;B; �] ; if and only if, there exist s1 (z) ; s2 (z) 2 S� [A;B; �] such that, for z 2 E;

f (z) =(s1 (z))

(m+24 )

(s2 (z))(m�24 )

; m � 2; z 2 E:

Proof. Since f (z) 2 Rm [A;B; �] if and only if zf0(z)f(z)

2 Pm [A;B; �] ; so we have

zf 0(z)

f(z)=

�m

4+1

2

�p1 (z)�

�m

4� 12

�p2 (z) ;

where p1 (z) ; p2 (z) 2 P [A;B; �] : Now with pi (z) = zs0i(z)

si(z); i = 1; 2; si 2 S�[A;B; �], we

havezf 0(z)

f(z)=

�m

4+1

2

�zs01(z)

s1(z)��m

4� 12

�zs02(z)

s2(z);

which on integration gives

log f (z) =

�m

4+1

2

�log s1(z)�

�m

4� 12

�log s2(z);

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