Some Classes of Generalized Convex and Related

Functions

A thesis submitted for the award of degree of

DOCTOR OF PHILOSPHY IN MATHEMATICS

By

WASIM UL-HAQ

Department of Mathematics

COMSATS Institute of Information Technology

Islamabad, Pakistan

August 2010

Some Classes of Generalized Convex and

Related Functions

A thesis presented to COMSATS Institute of Information Technology, Islamabad in partial fulfillment of the

requirement for the degree of

DOCTOR OF PHILOSPHY IN MATHEMATICS

By

Wasim Ul-Haq

CIIT/FA07-PMT-004/ISB

COMSATS Institute of Information Technology,

Islamabad, Pakistan

i

Some Classes of Generalized Convex and

Related Functions

A Post Graduate thesis submitted to the Department of Mathematics

As partial fulfillment for the award of Degree

DOCTOR OF PHILOSPHY IN MATHEMATICS

Name Registration Number

Wasim Ul-Haq CIIT/FA07-PMT-004/ISB

Signature:………..

Wasim Ul-Haq CIIT/FA07-PMT-004/ISB

Supervisor:…………….. Head of Department:…..… Prof. Dr. Khalida Inayat Noor Dr. Moiz-ud-Din Khan Professor Associate Professor Department of Mathematics Department of Mathematics CIIT, Islamabad CIIT, Islamabad

COMSATS Institute of Information Technology, Islamabad, Pakistan

ii

Final Approval

This thesis titled Some Classes of Generalized Convex and

Related Functions

Submitted for the degree of

DOCTOR OF PHILOSPHY IN MATHEMATICS

By

Wasim Ul-Haq

CIIT/FA07-PMT-004/ISB

has been approved for COMSATS Institute of Information Technology, Islamabad

External Examiner 1: __________________________ Dr. M. Yousaf Malik

Assistant Professor, Department of Mathematics QAU, Islamabad

External Examiner 2: __________________________ Dr. Siraj-ul-Islam

Associate Professor, Department of Basic Sciences UET, Peshawar

Supervisor: __________________________

Prof. Dr. Khalida Inayat Noor Professor, CIIT, Islamabad

Head, Department of Mathematics: __________________________ Dr. Moiz-Ud-Din Khan

Associate Professor, CIIT, Islamabad

Dean, Faculty of Sciences: __________________________ Prof. Dr. Arshad Saleem Bhatti

Professor, CIIT, Islamabad

iii

Certificate

I hereby declare that this thesis neither as a whole nor as a part there of has been copied out from any source. It is further declared that I have developed this thesis on the basis of my personal efforts made under the sincere guidance of my supervisor. No portion of the work presented in this thesis has been submitted in support of any other degree or qualification of this or any other University or Institute of learning, if found I shall stand responsible.

Signature:_______________

Name: Wasim Ul-Haq

Registration No: CIIT/FA07-PMT-004/ISB

iv

DEDICATED

To

My parents and wife For their prayers and love

&

To

Aiman & Abdullah

v

Acknowledgements

I am thankful to Allah Almighty, the Most Merciful and Compassionate, the Most Gracious and Beneficent, Whose bounteous blessings enabled me to perceive and pursue higher ideals of life, Who has given me the abilities to do the sheer hard work and enthusiasm to perform well. I gratefully acknowledge my supervisor Prof. Dr. Khalida Inayat Noor, Professor, Department of Mathematics, CIIT, Islamabad, under whose supervision, guidance and illustrative advice, the research work presented in this thesis became possible. She was always available to offer her guidance and encouragement to me. I express heartfelt and highly indebted gratitude to my teacher Professor Dr. Muhammad Aslam Noor, Professor, Department of Mathematics, CIIT, Islamabad, for his sincere advice, skilful guidance and valuable suggestions during difficult phases of my research. I express my deepest gratitude to Honorable Rector, Dr. S. M. Junaid Zaidi, CIIT, Pakistan, for providing ideal atmosphere of study and research in the department. I am also grateful to Head, Department of Mathematics, CIIT, Islamabad for providing all necessary facilities and research environment in the department. The role of Higher Education Commission of Pakistan in promoting a research culture and harnessing a knowledge revolution is highly commendable. I have strong feelings of appreciation for the Higher Education Commission of Pakistan, for financial support in the form of scholarship under Indigenous Ph.D Fellowship Program Batch-IV and for providing the latest literature in the form of the updated digital and reference libraries. This study would have been impossible, without the prayers, love, help, encouragement and moral support of my family. I express my appreciation and deep sense of gratitude from the core of my heart to my parents and wife whose hands always arise in prayer for my success. Words of gratitude and appreciation don’t always convey the depth of one’s feelings but I wish to thank my friends and colleagues who really helped me and kept my moral high during my thesis. Wasim Ul-Haq August, 2010

vi

Abstract

In this thesis, certain classes of analytic functions, such as ,

,

( , )UKηκ λ− α

− 1 2( , , )kR( , )UQCηκ λ α , ( , , )kN η ρ β , ( , )kR m ρ and , ( , , , , )pkR a b c A Bλ are

introduced. These classes generalize the concepts of uniformly close-to-convex and

quasi-convex, bounded turning, strongly close-to-convex, bounded boundary rotations

and bounded radius rotations. These classes are special generalizations of convex and

related functions. The techniques of convolution and differential subordination are

employed to investigate certain problems such as inclusion results, radius problems, arc

lengths, growth rate of coefficients and Hankel determinant problems and other several

interesting properties of the above mentioned classes. Some well-known results appear as

special cases from our main results.

γ γ β

vii

List of Symbols A Class of normalized analytic functions in the open unit disk E A ( Class of normalized p-valent functions in the open unit disk E )p

Set of complex numbers C Class of convex functions

( )C ρ Class of convex functions of order ρ

( )C β Class of strongly convex functions of order β

( )C η Class of convex functions of complex order η

*C Class of quasi-convex functions

mD Ruscheweyh derivative of m-th order Dα

λ Fractional calculus operator E Open unit disk 2 1 ( , , ; )F a b c z Gauss Hypergeometric functions

mI Noor integral operator of m-th order

viii

( )k z Koebe function K Class of close-to-convex functions

( )K ρ Class of close-to-convex functions of order ρ

( )K β Class of strongly close-to-convex functions of order β

( )M γ Class of γ -convex functions

Set of natural numbers P Class of functions with positive real part

( )P ρ Class of functions with positive real part greater than ρ

kR Class of bounded radius rotation

( )kR ρ Class of bounded radius rotations of order ρ S Class of normalized univalent functions

*S Class of starlike functions

* ( )S ρ Class of starlike functions of order ρ

* ( )S β Class of strongly starlike functions of order β

*Sη Class of starlike functions of complex order η UCV Class of uniformly convex functions UK Class of uniformly close-to-convex functions UST Class of uniformly starlike functions

UCVκ − Class of κ - uniformly convex functions

ix

UKκ − Class of κ - uniformly close-to-convex functions

kV Class of bounded boundary rotations

( )kV ρ Class of bounded boundary rotations of order ρ ( )nx Pochhammer symbol

( , ; )a c zφ Incomplete beta function ( * )( )f g z Convolution of ( ) and ( )f z g z

Subordination O(1) Constant depends on different parameters

x

Introduction

Riemann made significant and major contributions to mathematics, particularly in the

field of complex analysis. He developed the geometric approach to complex analysis

based on the Cauchy-Riemann equations and conformal mappings. His work with its

quality and insight gave birth to fascinating area of mathematics called geometric

function theory of a complex variable. Geometric function theory deals with the

geometric properties of the analytic functions. Riemann proved that there always exists

an analytic function that maps conformally any given simply connected domain

with at least two boundary points onto the open unit disk [19, 124]. The

Bieberbach’s conjecture which remained open for a long time has been positively settled

by de- Branges in the year 1984. He used hypergeometric functions in proving this open

problem.

( )f z

D ≠

Geometric function theory have recently found many applications in the fields of applied

sciences such as engineering, physics, electronics, signal theory and other branches of

applied sciences, for some details, see [124]. Yet it is continued to find new applications

in other fields of sciences such as theory of partial differential equations, fluid dynamics,

modern mathematical physics and non-linear integrable systems.

In this area of complex analysis, we mainly deal with the class S of functions which are

univalent in the open unit disk :| | 1E z z= < . Such functions were first studied by

Koebe in 1907. Later, Bieberbach's proof of second coefficient estimate of normalized

univalent functions in 1916 is the corner stone of this field. On the basis of Bieberbach's

theorem and Koebe function, he enabled to conjecture the famous result regarding the

coefficients estimate of the univalent functions which stood as a challenging problem of

the field and attracted the attention of many mathematicians until de Branges proved it in

1984, for some details, see [19, 28, 29, 49, 113, 117, 124].

The geometry theory of functions of single-valued complex variable is mostly concerned

with the study of the properties of univalent functions. The image domain of E under a

xi

univalent function is of interest if it has some nice geometric properties. A convex

domain is an outstanding example of a domain with nice properties. Another example of

such domain is star- shaped with respect to a point. Certain subclasses of those analytic

univalent functions which map E onto these geometric figures, are introduced and their

properties are widely investigated, for example the classes C and of convex and

starlike functions respectively, for detail see [19, 28]. Moreover, the classes of convex

and starlike functions are closely related with the class P of analytic functions with

and , see [19, 28].

*S

( )h z

(0) 1h = Re ( ) 0h z >

In 1952, Kaplan studied the class K of univalent functions which maps E onto domain with some interesting geometric meaning. He named such functions as close-to-convex functions. It was proved that “Each close-to-convex function is univalent”. Geometrically, means that maps each circle | |( )f z K∈ ( )f z 1z r≤ < into a simple closed curve whose tangent rotates, as θ increases, either in a clockwise or anti-clockwise direction in such a way that it never turns back onto itself as much as to completely reverse its direction, see [28, vol.2]. Noor [59], Noor and Thomas [83], investigated a new class of univalent functions which is closely related to close-to-convex functions and have many interesting properties. This is known as the class of quasi- convex functions and it is denoted by . It was observed that both of these classes are related with each other through classical Alexander type relation. That is

*C

*( )f z C∈ ( ) .zf z K′<==> ∈

The natural generalization of convex functions was provided by Goodman [26, 27] in

1991, by introducing the concepts of uniformly convex and uniformly starlike functions.

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

A function A is called uniformly convex (starlike) in E if is in and

for every circular arc in E, with center

( )f z ∈ ( )f z ( )C S ∗

ϒ ξ , also in E, the arc is also convex

(starlike with respect to

( )f ϒ

( )f ξ ). Following the notations of Goodman [26, 27], we denote

by UCV and UST, the classes of uniformly convex and uniformly starlike functions

respectively.

Later Ronning [108] (independently Ma and Minda [46]) and several other obtained a

most suitable form of Goodman criteria of uniformly convex, which are related to conic

xii

regions. The classic Alexander's theorem stating that ( )f z C∈ if and only if *( )zf z S′ ∈

provides a bridge between these two classes. One might hope that there would be a

similar bridge between UCV and UST, but two examples in [27] show that this is not the

case, that is, if and only if ( )f z UCV∈ ( )zf z UST′ ∈ failed. The class

( ) : ( ) ( ), ( )ST g z g z zf z f z UCV′= = ∈

was introduced by Ronning [108] to verify whether ST is a proper subclass of UST or not.

Later he proved (see [113]) that neither ST UST⊄ nor UST ST⊄ . Ronning [108], and

Ma and Minda independently gave a more applicable one variable characterization of

the class UCV . He observed that functions in the class ST maps E onto parabolic region.

This idea was extended to conic regions in general by Kanas and Winiowska [34, 35].

Later Kumar, Ramesha [38] and Subermanian et. al. [120] studied the classes of close-to-

convex and quasi-convex functions associated with the parabolic regions.

Denote by the set of all functions kV2

( ) nn

nf z z a z

∞

=

= +∑ which are analytic in the unit

disk E and map E onto a domain with boundary rotation at most kπ and it is

geometrically clear that . When 2k ≥ 2k = , is the class of normalized convex

functions. It is known [28], for

2V C

2 4, kk V K S⊂ . The functions in can be

represented by a Stieljes integral where the integrator

kV

( )tµ is of bounded variation on

[ ],π π− , and the total increase of ( )tµ is 2π and the total variation of ( )tµ on [ ],π π−

is at most kπ . A natural extension of the class of convex functions is the class .

This class was first introduced by Löwner [45] in 1917, but Paatero [89, 90] developed

the basic notation and theory of this class. Brannan [9] and Brannan et.al [10], Noor [63,

69, 78, 79, 80, 81] and many more investigated various aspects and applications of the

class . Geometrically f function

2,k kV ≥

2,k kV ≥ ( ) kf z V∈ means that it maps E conformally onto

a domain whose boundary rotation is at most kπ . In the same way Tammi [121], in 1952

introduced the class kR of bounded radius rotations by extending the idea of starlike

functions.

≤ ≤ ⊂

xiii

In chapter 1, we give some elementary concepts from geometric function theory which is

used in the upcoming chapters. In chapter 2, using fractional calculus operator we

introduce and discuss some classes, ( , ) and ( , )UK UQη ηκ λ α κ λ α− − for

and 0, 0, 0 1, 0κ λ α η≥ ≥ ≤ < ∈ − z E∈ . The contents of this chapter have been

published in the journal of Applied Mathematics and Computation [85]. With different

choices of parameters a number of inclusion results for the classes

are investigated. The convolution of these classes with

convex functions along with some applications is also given. The coefficient bounds for

these classes are also obtained.

( , ) and ( , )UK UQηκ λ α κ λ− − η α

In chapter 3, we introduce the class 1 2( , , )kR γ γ β by which generalizes the class of

functions with bounded turning. Some interesting properties such as inclusion results,

integral preserving property and radius problem for these functions are discussed. The

contents of this chapter are already published in the journal of Non-Linear Functional

Analysis and Applications [84].

In chapter 4, we introduce the class ( , , )kN η ρ β , which generalizes the class of

functions strongly close-to-convex functions. Some interesting properties such as radius

of convexity problem, arc length and coefficient growth problem for these functions are

discussed. The growth rate of Hankel determinant for this class is also be our point of

investigation.

Using the terminology of order, Padmanabhan and Parvatham [91], generalized the

classes of bounded boundary and bounded radius rotations. A detailed discussion on

these classes is given in chapter one. In chapter 5, we study certain classes of analytic

functions defined by Ruscheweyh derivatives. Some basic properties involving

generalized Bernadi integral transform, inclusion results and a radius problem are

investigated. Many interesting special cases of these results are also observed.

In chapter 6, we introduce a new subclass of analytic functions by using

the generalized integral operator defined in terms of convolution with hypergeometric

functions [50]. The class generalizes the class of bounded boundary and

, ( , , , A,B)pkR a b cλ

, ( , , , A,B)pkR a b cλ

xiv

bounded radius rotations of order ρ . Many interesting inclusion relationships and radii

problems are investigated regarding this class. In this chapter, we are mainly focused on

presenting some generalization and applications of the class of p-valent functions. A part

of this work is published in the Journal of Inequalities and Applications [86]. We also

observe that this class is preserved under the Bernardi integral operator by varying

different order.

xv

Contents

1 Elementary concepts from geometric function theory 3

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Analytic and univalent functions . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Functions with positive real part and related classes . . . . . . . . . . . . 6

1.4 Some basic subclasses of univalent functions . . . . . . . . . . . . . . . . 13

1.4.1 Starlike and convex functions . . . . . . . . . . . . . . . . . . . . 13

1.4.2 The class of close-to-convex functions . . . . . . . . . . . . . . . . 20

1.4.3 Uniformly convex, uniformly starlike and related functions . . . . 22

1.5 The class of bounded boundary rotation and related topics . . . . . . . . 26

1.5.1 Functions with bounded boundary rotation . . . . . . . . . . . . . 26

1.5.2 Functions with bounded radius rotation . . . . . . . . . . . . . . . 27

1.5.3 Some related classes with bounded boundary rotation . . . . . . . 29

1.6 Differential subordination . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.6.1 Hypergeometric functions . . . . . . . . . . . . . . . . . . . . . . 35

1.7 Convolution (Hadamard product) and certain linear operators . . . . . . 37

1.7.1 Certain linear operators defined in terms of convolution . . . . . . 39

2 On classes of κ-uniformly close-to-convex and related functions 44

2.1 The class of κ-uniformly close-to-convex functions of complex order . . . 45

2.1.1 Some properties of the class κ− STη(λ, α) and κ− UKη(λ, α) . . 49

2.1.2 Convolution invariance with convex function . . . . . . . . . . . . 55

1

2.1.3 Sufficient condition for functions in κ− UKη(λ, α) . . . . . . . . . 58

2.2 The class κ− UQCη(λ, α) . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.2.1 Inclusion results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.2.2 Coefficient Estimates . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.3 Some applications of convolution invariance . . . . . . . . . . . . . . . . 62

3 Some properties of a subclass of analytic functions 63

3.1 Certain properties of the class]Rk(γ1, γ2, β) . . . . . . . . . . . . . . . . . 64

4 On a certain class of analytic functions and Hankel determinant prob-

lem 70

4.1 Some properties of the class]Nk(η, ρ, β) . . . . . . . . . . . . . . . . . . . 72

4.2 Hankel determinant problem . . . . . . . . . . . . . . . . . . . . . . . . . 77

5 Some applications of Ruscheweyh derivatrives 86

5.1 Certain analytic functions defined by Rusheweyh derivatives . . . . . . . 87

5.1.1 Some inclusion problems and integral preserving property . . . . . 87

5.1.2 Various interesting implications . . . . . . . . . . . . . . . . . . . 93

5.1.3 Radius problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6 On certain class of p-valent functions defined by some integral operator 96

6.1 An integral operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.2 A class of analytic p-valent functions . . . . . . . . . . . . . . . . . . . . 99

6.3 Some inclusion results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

2

Chapter 1

Elementary concepts from geometric

function theory

1.1 Introduction

This chapter is of introductory nature. In this chapter, we give a brief introduction of

elementary concepts from geometric function theory, which will be useful in our later

chapters. This chapter comprises seven sections. The proofs of most of the results can

be seen in the standard text books, for example see [19, 28, 100].

In the second section of this chapter, we give the concepts related to analytic and univa-

lent functions which play a key role in our work.

The third section deals with the class of functions with positive real part and its related

subclasses. Various interesting properties of these classes are given for completeness pur-

pose.

In the fourth section, some basic subclasses of univalent functions are given, which are

defined by analytic and geometric conditions. These include the well-known classes of

starlike and convex functions, the classes of close-to-convex and quasi-convex functions

and the classes of uniformly convex and uniformly starlike functions.

The fifth section is mostly concerned with the classes of functions with bounded boundary

3

and bounded radius rotations. We also discuss their analytic and geometric characteri-

zations along with their relationships to different subclasses of analytic functions.

In the sixth and seventh sections, the concepts of differential subordination and con-

volution are discussed which are very effective tools in our later investigations. These

sections include the hypergeometric functions and a survey on different linear operators.

Hypergeometric functions were first used by Carlson and Shaffer [13] and in the proof of

Bieberbach conjecture by de Branges [8]. After this, these functions are frequently used

in geometric function theory. We also present some basic operators including general-

ized Bernardi integral operator, Ruscheweyh derivative operator, Noor integral operator,

Carlson and Shaffer operator and Fractional calculus operator.

1.2 Analytic and univalent functions

In this section, we shall discuss briefly the class A of normalized analytic functions. The

main aim is to introduce the class S of normalized univalent functions defined in the

open unit disk E = z ∈ C : |z| < 1.

The class A of normalized analytic functions [19, 28, 100]

A complex-valued function f (z) of the complex variable z is said to be differentiable at

a point z0 in C, if limz−→z0

f(z)− f(z0)

z − z0exists. Such a function f(z) is analytic at z0 if it

is differentiable at z0 and at every point in some neighborhood of z0. A complex-valued

function f(z) of the complex variable z is said to be analytic in a domain D, if it is

analytic at every point in D.

It is one of the miracles of complex analysis that an analytic function f(z) must have

derivatives of all orders and that f(z) has Taylor series representation

f(z) =∞Xn=0

an(z − z0)n, an =

f (n)(z0)

n!

4

convergent in some open disk at z0.We shall be mostly concerned primarily with the class

A of functions which are analytic in the open unit disk E = z : |z| < 1 , normalized by

the conditions f(0) = 0 and f 0(0) = 1. These normalization conditions do not effect the

generality. Thus each such function has the form

f (z) = z +∞Xn=2

an zn, z ∈ E. (1.2.1)

The selection of open unit disk E above instead of an arbitrary domain D is due to

Riemann Mapping Theorem (proof can be found in standard text for example, see [19])

which says that any simply connected domain D ⊂ C with at least two boundary points

can be mapped conformally onto the open unit disk E.

Univalent function [19, 28, 29, 100]

A function f (z) analytic in E is said to be univalent in E, if w = f(z) assumes distinct

values w for distinct z in E. In this case the equation f(z) = w has at most one root

in E. Other terminology such as simple, or Schlicht (German word for simple) are also

used for univalent. In more precise way, we can say that a univalent function is one, that

never takes the same value twice; that is, f (z1) 6= f (z2) for all points z1 and z2 in E with

z1 6= z2. In simple language, a univalent function f(z) is one-to-one (injective) mapping

of E onto another domain.

We denote by S the class of all those functions which are univalent and of the form

(1.2.1). The foremost example of such functions is the Koebe function denoted by k(z)

and given as

k(z) =1

4

µ1 + z

1− z

¶2− 14=

z

(1− z)2= z +

∞Xn=2

n zn, z ∈ E. (1.2.2)

The two basic results due to Bieberbach (1916), stating that when f(z) ∈ S, then |a2| ≤ 2

and f(z) assumes every value w such that |w| < 14. Both of these results are sharp.

5

Finally it can be noted that the Koebe function k(z) maps the disk E, one-one and

conformally onto the w−plane cutting from−14to−∞ (minus infinity) along the negative

real axis. This Koebe function k(z) plays a central role in the class S of normalized

univalent functions due to its extremal nature. In 1985, de Branges settled the famous

problem in the univalent function theory, by proving the Bieberbach conjecture for the

coefficient estimates of the class S that |an| ≤ n holds for n ≥ 2.

The term locally univalent is also often used in the literature. We define it as below:

Locally univalent function [19, 28, 100]

A function f(z) is called locally univalent at a point z0 ∈ E, if it is univalent in some

neighborhood of z0. For analytic function f(z) the condition f 0(z0) 6= 0 is equivalent to

local univalence at z0. Locally univalent functions are also called conformal (angle and

sense preserving mapping).

The next section is about some important functions, so called the functions with

positive real part. We shall see that most of the subclasses of univalent functions are

directly related to the class of functions with positive real part. They also play a key

role in a variety of problems from geometric function theory of a complex variable and

related fields.

1.3 Functions with positive real part and related classes

Functions which map the open unit disk E onto right half plane are of particular interest.

They, for instance, play a vital role in geometric function theory. They are related to

univalent functions in one way or the other. Many problems are solved by using the

properties of these functions. In this section, we study the class P , consisting of all the

functions which has positive real part. Some related classes will also be introduced and

some of their basic properties are given. These properties will be very useful in our later

investigations, see [28].

6

Let P denotes the class of all functions h(z) which are analytic in E with

h(0) = 1, Reh(z) > 0, z ∈ E.

It can be noted that these functions are represented by the following series

h (z) = 1 +∞Xn=1

cn zn, z ∈ E. (1.3.1)

Relationship with univalent functions

The functions in the class P need not to be univalent and vice versa. For example, the

function

h (z) = 1 + zn ∈ P for any integer n ≥ 0,

but if n ≥ 2, this function is no longer to be univalent. The Möbius function

m0(z) =1 + z

1− z, z ∈ E,

is analytic and univalent in E. Moreover, it maps E onto the right half plane, for details,

see [19, 28].

The role of Möbius function m0(z) is same as that of Koebe function in the class S. The

functionm0(z) is not the only function with extreme properties for the class P. There are

many other functions of the form (1.3.1), which are extreme for the class P. No one of

these functions is achievable from others. In 1915, Alexandar proved an interesting result

for the univalence of analytic functions, see [28]. He showed that, if Re f 0(z) > 0 for each

z ∈ E, then f(z) is univalent in E. Furthermore, in 1935, Noshiro [57] and Warschawski

[122] independently proved that, if for a function f(z) analytic in a convex domain D,

Re eiαf 0(z) > 0, α real, z ∈ D,

then f(z) is univalent in that domain.

7

We observe a very useful fact that the class P forms a convex set. By this we mean

that if µ1 and µ2 are non-negative numbers with µ1 + µ2 = 1, h1(z) and h2(z) are in P ,

then

h(z) = µ1h1(z) + µ2h2(z), (1.3.2)

is also in P . It is obvious that (1.3.2) holds for µj ≥ 0 such thatnX

j=1

µj = 1 and hj(z) is

in the class P . We also see that (1.3.2) holds for infinite convex combination. That is

h(z) =∞Xj=1

µjhj(z), (1.3.3)

if we assume that µj ≥ 0 for each j, and∞Xj=1

µj = 1. We can replace (1.3.3) by a

Stieltjes integral, if the functions hj(z) are properly selected. If hj(z) is in the class P ,

then for each real τ , hj(e−iτz) is in the class P . We apply this to the function m0 (z)

and extend (1.3.3) to a Stieltjes integral in which the positive weight µj is replaced by

dµ(τ). The upcoming theorem is a representation formula due to Herglotz (1911) and a

characterization of coefficients.

Theorem 1.3.1 [100]

Let h(z) be analytic in E and of the form (1.3.1). Then the following statements are

equivalent:

(i) The function h(z) is in the class P ;

(ii) there exists a non-decreasing function µ (τ) (0 ≤ τ ≤ 2π) such that

h (z) =1

2π

2πZ0

m0

¡e−itz

¢dµ (τ) , µ (2π)− µ (0) = 2π; (1.3.4)

8

(iii) for ν = 1, 2, 3, . . . ,

νXj=0

νXl=0

cj−lλjλl ≥ 0, (λ0, . . . λν ∈ C), (1.3.5)

where the convention c0 = 2, c−j = cj (j ≥ 1) is adopted. ¤In the results below, we give the coefficient bounds, growth and distortion results for the

class P .

Corollary 1.3.1 [100]

Let h(z) ∈ P and of the form (1.3.1) . Then

|cn| ≤ 2, for n = 1, 2, 3, . . . ,

and the inequality is sharp, if and only if

h(z) =nX

υ=1

µυeiα+2πiυ/n + z

eiα+2πiυ/n − z

for some α and µ1, . . . , µn ≥ 0 and µ1 + . . .+ µn = 1. ¤

Theorem 1.3.2 [28]

Let h(z) ∈ P. Then for |z| = r < 1

1− r

1 + r≤ Reh(z) ≤ |h(z)| ≤ 1 + r

1− r, (1.3.6)

and

|h0(z)| ≤ 2Reh(z)1− r2

. (1.3.7)

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

the Möbius function m0(z). ¤

9

Lemma 1.3.1 [30]

Let h(z) ∈ P with z = reiθ. Then

2πZ0

¯h(reiθ)

¯λdθ < C (λ)

1

(1− r)λ−1, (1.3.8)

where λ > 1 and C (λ) is a constant depending on λ only. ¤

Lemma 1.3.2[98]

Let h(z) ∈ P with z = reiθ. Then

1

2π

2πZ0

|h(z)|2 dθ ≤ 1 + 3r2

1− r2.

By using the order terminology (Robertson, 1936), we replace the condition Reh(z) > 0

by

Reh(z) > ρ (0 ≤ ρ < 1, z ∈ E). (1.3.9)

Let P (ρ) denotes the class of all functions which satisfy (1.3.9). Note that P (0) = P, the

following relation between the classes P (ρ) and P can easily be observed

h(z) = (1− ρ)h1(z) + ρ, h1(z) ∈ P. (1.3.10)

From (1.3.4) and (1.3.10), we obtain the following Herglotz formula for the class P (ρ)

h(z) =1

2π

2πZ0

1 + (1− 2ρ) ze−it1− ze−it

dµ(τ), µ(2π)− µ(0) = 2π. (1.3.11)

10

The class Pk(ρ)[91]

Let Pk(ρ) be the class of functions h(z) with h(0) = 1, which are analytic in E and

satisfying2πZ0

¯Re

½h(z)− ρ

1− ρ

¾¯dθ ≤ kπ, (1.3.12)

where z = reiθ, k ≥ 2 and 0 ≤ ρ < 1. This class has been investigated by Padmanabhan

and Parvatham [91]. For ρ = 0, we obtain the class Pk, introduced by Pinchuk [93] and

for ρ = 0, k = 2, we obtain the class P of functions with positive real part.

The Herglotz illustration formula for the functions in class Pk(ρ) is as follows

h(z) =1

2π

2πZ0

1 + (1− 2ρ) ze−it1− ze−it

dµ(τ), (1.3.13)

where µ(τ) is a function with bounded variation on [0, 2π] such that for k ≥ 2,

µ(2π)− µ(0) = 2π and

2πZ0

|dµ(τ)| ≤ kπ. (1.3.14)

Since the integrator µ(τ) has a bounded variation on [0, 2π], we may write

µ(τ) = Y1(τ)− Y2(τ),

where Y1(τ) and Y2(τ) are two non-negative increasing functions on [0, 2π] satisfying

(1.3.13). Thus, if we take

Y1(τ) =

µk

4+1

2

¶µ1(τ) and Y2(τ) =

µk

4− 12

¶µ2(τ),

11

then (1.3.13) becomes

h (z) =

µk

4+1

2

¶1

2π

2πZ0

1 + (1− 2ρ)ze−iτ1− ze−iτ

dµ1(τ)

−µk

4− 12

¶1

2π

2πZ0

1 + (1− 2ρ)ze−iτ1− ze−iτ

dµ2(τ).

Now, using Herglotz-Stieltjes formula for function in the class P (ρ) given in (1.3.11), we

obtain

h (z) =

µk

4+1

2

¶h1 (z)−

µk

4− 12

¶h2 (z) , z ∈ E, (1.3.15)

where hi (z) ∈ P (ρ) for i = 1, 2. It is known [74], that Pk(ρ) forms a convex set.

Theorem 1.3.3 [93]

Let h(z) ∈ Pk. Then1− kr + r2

1− r2≤ Reh(z) ≤ 1 + kr + r2

1− r2, (1.3.16)

for all z ∈ E with |z| = r and k ≥ 2. ¤Proof. Let h(z) ∈ Pk. Then, using (1.3.15), we can write

Reh (z) =

µk

4+1

2

¶Reh1 (z)−

µk

4− 12

¶Reh2 (z) , z ∈ E,

where hi(z) ∈ P for i = 1, 2. Using (1.3.6), we obtain

Reh (z) ≤µk

4+1

2

¶1 + r

1− r−µk

4− 12

¶1− r

1 + r

=1 + kr + r2

1− r2.

The left hand side of (1.3.16) can be proved in similar manner as above. ¥The class P is directly related through their derivatives to a number of important and

12

basic subclasses of univalent functions (e.g., convex and starlike). In the next section,

we define some subclasses of univalent functions along with their geometric properties.

1.4 Some basic subclasses of univalent functions

Geometric function theory of a single-valued complex variable is mostly concerned with

the study of the properties of univalent functions. The image domain of E under a

univalent function is of interest if it has some nice geometric properties. A convex domain

is an outstanding example of a domain with interesting properties. Another example of

such domain is star-shaped with respect to a point. Here we briefly discuss some basic

subclasses of these analytic univalent functions which mapE onto these geometric figures.

We will also be interested in finding their basic properties, relationships with each other

and many other results.

1.4.1 Starlike and convex functions

1. The function f(z) of the form (1.2.1) is known to be starlike in E if it is univalent and

if the image domain D = f(E) is starshaped with respect to 0, that is

w ∈ D, 0 ≤ t ≤ 1 =⇒ tw ∈ D.

Now we give analytic characterization for such functions. A function f(z) from the class

A is starlike in E, if and only if

Rezf 0(z)

f(z)> 0 or

zf 0(z)

f(z)∈ P, for z ∈ E. (1.4.1)

For the proof of this characterization, we refer to [19, 28, 100].

We denote by S∗ the set of all functions which are starlike in E. This class was first

investigated by Alexander and the analytic characterization (1.4.1) is due to Nevanlinna,

see[19, 28, 100]. We now derive a representation formula for the class S∗ of starlike

13

functions.

Theorem 1.4.1

A function f(z) ∈ A is starlike in E, if and only if

f(z) = z exp

⎡⎣2 2πZ0

log1

1− e−iτzdµ(τ)

⎤⎦ , z ∈ E, (1.4.2)

for some increasing function µ(τ) with µ(2π)− µ(0) = 1. ¤Proof. Let f(z) be starlike. From Theorem 1.3.1 and condition (1.4.1), we have

zf 0(z)

f(z)=

2πZ0

1 + e−iτz

1− e−iτzdµ(τ),

where µ(τ) has the desired properties. Hence

f 0(z)

f(z)− 1

z=

2πZ0

2e−iτ

1− e−iτdµ(τ).

Further integration gives us

logf(z)

z− log f 0(0) = −2

2πZ0

log(1− e−iτz)dµ(τ),

which implies (1.4.2). The converse case is obtained just by reversing this argument. ¥

Examples 1.4.1

Let αj > 0 (j = 1, 2, . . . , n) and α1 + α2 + . . .+ αn = 2. Then

f(z) = znY

j=1

¡1− e−iθz

¢−αj, z ∈ E

14

is starlike in E. This follows from the above theorem by selecting the µ(τ) as a step

function with jumps 12αj at the points θj (j = 1, 2, . . . n). The Koebe function is a

special case. By considering f(eiθ), we observe that the image domain is the plane minus

n slits.

2. The function f(z) of the form (1.2.1) is called convex in E, if it is univalent and if the

image domain D = f(E) is convex. That is

For w1, w2 ∈ D (0 ≤ t ≤ 1) =⇒ (1− t)w1 + tw2 ∈ D.

In simple language a convex domain is one that is starshaped with respect to each of its

points. The analytic characterization for convex function is given by

Re

∙1 +

zf 00(z)

f 0(z)

¸> 0, z ∈ E. (1.4.3)

The class of all functions which are convex in E is denoted by C. For instance, the

functionsz

1− zand log

∙1 + z

1− z

¸are convex in E. From the above discussion, we observe that

C ⊂ S∗ ⊂ S.

The Koebe function (1.2.2) is starlike but not convex, see [28]. The following beautiful

relation between C and S∗ due to Alexander (1915) can easily be seen and is given as:

f(z) ∈ C ⇐⇒ zf 0(z) ∈ S∗, z ∈ E. (1.4.4)

15

This relation in equivalence form can also be revealed as:

If F (z) ∈ S∗, then the integral

A(F (z)) =

zZ0

F (τ)

τdτ (1.4.5)

is in the class C. The integral (1.4.5) is known as Alexander integral operator in the

literature.

We now give the sharp coeffiecient estimates for starlike functions [28].

Theorem 1.4.2 (Nevanlinna,1921)

If f(z) is in the class S∗ and of the form (1.2.1), then

|an| ≤ n, for each n ≥ 2. (1.4.6)

Further, the sharpness of this inequality for each n can be viewed from Koebe function

or one of its rotation. ¤The following result is an immediate consequence from the above theorem.

Corollary 1.4.1

If f(z) is in the class C and of the form (1.2.1), then

|an| ≤ 1, for each n ≥ 2. (1.4.7)

Further, the sharpness can be seen from the function

l(z) =z

1− z, z ∈ E. ¤ (1.4.8)

The sharp distortion and growth results for starlike functions and convex functions are

stated as follows

16

Theorem 1.4.3[28]

If f(z) ∈ S∗, then for |z| = r < 1,

r

(1 + r)2≤ |f(z)| ≤ r

(1− r)2,

and1− r

(1 + r)3≤ |f 0(z)| ≤ 1 + r

(1− r)3.

Equalities occur if f(z) is a suitable rotation of the function k(z) given by (1.2.2). ¤Using the relation (1.4.4), we have the following.

Corollary 1.4.2[28]

If f(z) ∈ C, then for |z| = r < 1,

r

(1 + r)≤ |f(z)| ≤ r

(1− r),

and1

(1 + r)2≤ |f 0(z)| ≤ 1

(1− r)2.

Equalities occur if f(z) is a suitable rotation of the function l(z) given by (1.4.8). ¤

Lemma 1.4.1 [24]

Let f(z) be univalent and 0 ≤ r < 1. Then there exists a number z1 with |z1| = r, such

that for all z, |z| = r, we have

|z − z1| |f(z)| ≤2r2

1− r2. ¤

Using the order terminology, Robertson [105] introduced the classes S∗(ρ) and C(ρ) of

starlike and convex functions of order ρ, 0 ≤ ρ < 1, which are defined by

17

S∗(ρ) =

½f(z) ∈ A : Re zf

0(z)

f(z)> ρ, z ∈ E

¾,

C(ρ) = f(z) ∈ A : zf 0(z) ∈ S∗(ρ), z ∈ E .

For ρ = 0, we obtain the well-known classes of starlike and convex univalent functions.

It is clear that S∗(ρ) ⊆ S∗ and C(ρ) ⊆ C. Strohhäcker [119], proved that each convex

function is starlike of order one half. The extension of this result is the following theorem.

Theorem 1.4.4

If 0 ≤ ρ < 1, then the order of starlikeness of convex functions of order ρ is given by

ρ1 = ρ1(ρ) =

⎧⎨⎩4ρ(1−2ρ)4−22ρ+1 , ρ 6= 1

2,

12 ln 2

, ρ = 12.

(1.4.9)

This result is sharp. ¤In 1971 Jack [32] gave the rough estimate for ρ1, while Goel [22] proved the exact

version of this result. Another proof with a different method can be found in [50].

The term complex order was first introduced by Nasr and Aouf [55]. They studied

the class S∗η of starlike functions of complex order η and Wiatrowski [123] considered the

class Cη of convex functions of complex order η which are defined as follows:

S∗η =

½f(z) ∈ A : Re

µ1 +

1

η

µzf 0(z)

f(z)− 1¶¶

> 0, z ∈ E

¾,

Cη =

½f(z) ∈ A : Re

µ1 +

1

η

zf 00(z)

f 0(z)

¶> 0, z ∈ E

¾,

where η ∈ C − 0. We note that for η = 1 − ρ, 0 ≤ ρ < 1, S∗η = S∗ (1− ρ) and

Cη = C (1− ρ).

The condition Reh(z) > ρ, for some analytic function h(z) with h(0) = 1, can be

interestingly altered by demanding |arg h(z)| ≤ βπ2, where 0 < β ≤ 1. We denote by

18

eP (β), the class of all functions satisfying the latter condition. Brannan, Kirwan andStankiewicz introduced the concept of strongly starlike and strongly convex functions,

see [28].

Strongly starlike and strongly convex functions

1. A function f(z) ∈ A is said to be strongly convex of order β in E, if for all z ∈ E,¯arg

½1 +

zf 00(z)

f 0(z)

¾¯<

βπ

2, 0 < β ≤ 1.

The class of such functions is denoted by eC(β).2. A function f(z) ∈ A is said to be strongly starlike of order ρ in E if for all z ∈ E,¯

arg

½zf 0(z)

f(z)

¾¯<

βπ

2, 0 < β ≤ 1.

The class of all such functions is denoted by fS∗(β).In 1969, Mocanu [54] introduced the concepts of γ-convex functions as:

The class of γ-convex functions

A function f(z) ∈ A is said to be γ-convex in E, iff(z)f 0(z)

z6= 0,

Re

½(1− γ)

zf 0(z)

f(z)+ γ

(zf 0(z))0

f 0(z)

¾> 0,

for all z in E. The class of all such functions is denoted by M(γ).

The above class is meaningful if we consider γ to be a complex number, but here we

assume γ to be real. Miller, Mocanu and Read [53] proved that all γ-convex functions are

convex if γ ≥ 1 and starlike if γ < 1. We note when γ = 1, then a γ-convex function is

convex and γ-convex function is starlike when γ = 0. Thus the setM(γ) gives continuous

transition from convex to starlike functions.

19

There is another interesting class of functions which has a simple geometric descrip-

tion. This is the class of close-to-convex functions, introduced by Kaplan [36] in 1952.

It was proved that close-to-convex functions are univalent, see [28]. Now, we will give a

brief introduction of these functions along with geometric illustration.

1.4.2 The class of close-to-convex functions

The function f(z) of the form (1.2.1), analytic in the unit disk E is said to be close-to-

convex, if there exists a starlike function g(z) such that

Re

½zf 0(z)

g(z)

¾> 0, for z ∈ E. (1.4.10)

We denote byK, the class of all functions that are close-to-convex. The name was chosen

because, by relation (1.4.4), inequality (1.4.10) is equivalent to

Re

½f 0(z)

h0(z)

¾> 0, for z ∈ E,

where h(z) is convex in E. Every convex function is obviously close-to-convex. More

generally, every starlike function is close-to-convex. It was proved in [36], that close-to-

convex functions are univalent. Therefore, it can be easily seen that C ⊂ S∗ ⊂ K ⊂ S.

Geometric description

Close-to-convex functions can be characterized by a simple geometric description. Let

f(z) ∈ A and let Cr be the image under f(z) of the circle |z| = r, where 0 < r < 1.

Roughly speaking, f(z) is close-to-convex, if and only if none of the curves Cr makes a

"reverse hairpin turn". More precisely, the requirement is that as θ increases, the tangent

direction arg©

∂∂θf(reiθ)

ªshould never decrease by as much as π from any previous value.

Since∂

∂θ

∙arg

½∂

∂θf(reiθ)

¾¸= Re

½1 +

zf 00(z)

f 0(z)

¾, z = reiθ,

the following theorem, due to Kaplan [36], can be stated as:

20

Theorem 1.4.5 (Kaplan’s Theorem)

Let f(z) be analytic and locally univalent in E. Then f(z) is close-to-convex, if and only

ifθ2Z

θ1

Re

½1 +

zf 00(z)

f 0(z)

¾dθ > −π, z = reiθ, (1.4.11)

for each r in (0, 1) and every pair θ1, θ2 with 0 ≤ θ1 < θ2 ≤ 2π. ¤Goodman [25], introduced and studied the class K (ρ) of close-to-convex functions of

order ρ, 0 ≤ ρ < 1, as follows.

Let f(z) be analytic and locally univalent in E. Then f(z) is close-to-convex of order ρ,

if and only if

θ2Zθ1

Re

½1 +

zf 00(z)

f 0(z)

¾dθ > −(1− ρ)π, z = reiθ, for 0 ≤ ρ < 1,

for each r in (0, 1) and every pair θ1, θ2 with 0 ≤ θ1 < θ2 ≤ 2π.

In a similar way as above, we can define strongly close-to-convex functions as:

A function f(z) ∈ A is said to be strongly close-to-convex of order β ≥ 0, in the open

unit disk E, if there exists a function g(z) ∈ C such that¯arg

½f 0(z)

g0(z)

¾¯<

βπ

2. (1.4.12)

We denote by eK(β) (0 < β ≤ 1) the class of all strongly close-to-convex functions of

order β. This class was first studied by Pommerenke [97]. For β = 1, eK(β) = K, but if

β < 1, then eK(β) ⊂ K, see [28].

The class of quasi-convex functions

Analogous to the class of close-to-convex functions, Noor [59] defined the class of quasi-

convex functions. It has the same relation with the class K of close-to-convex functions

as C has with S∗. The function f(z) of the form (1.2.1), analytic in the unit disk E is

21

said to be quasi-convex, if there exists a function g(z) in C, such that

Re

½(zf 0(z))0

g0(z)

¾> 0, for z ∈ E.

The class of all such functions is denoted by C∗. It was proved in [59, 83] that "Every

quasi-convex function is close-to-convex and hence univalent". Thus

C ⊂ C∗ ⊂ K ⊂ S.

Further it was observed that C∗ has no inclusion relationship with S∗, for more details

see [59, 62, 83].

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

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

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

1.4.3 Uniformly convex, uniformly starlike and related func-

tions

1. Let f(z) ∈ A. Then f(z) is called uniformly convex (uniformly starlike) in E, if f(z)

is in the class C (S∗) and for every circular arc Υ contained in E with center ζ also in

E, the arc f(Υ) is convex (starlike with respect to f(ζ)).

These classes were introduced by Goodman [26, 27] and following the notation of

Goodman, we denote by UCV and UST , the classes of uniformly convex and uniformly

starlike functions respectively. An analytic description of UCV and UST can be found

in [26, 27] . We state it in the following theorem.

22

Theorem 1.4.6

Let f(z) ∈ A. Then

(i) f(z) ∈ UCV, if and only if

Re

½1 + (z − ζ)

f 00(z)

f 0(z)

¾> 0, (z, ζ) ∈ E ×E, (1.4.13)

(ii) f(z) ∈ UST, if and only if

Re

½(z − ζ) f 0(z)

f(z)− f(ζ)

¾> 0, (z, ζ) ∈ E ×E. ¤ (1.4.14)

It is important to note that when we take ζ = 0 in (1.4.13) and (1.4.14), we obtain

the well known classes C and S∗ respectively. One might hope that there would be a

similar bridge between UCV and UST as between C and S∗, but two examples in [26]

show that this is not the case, that is, Alexander type result f(z) ∈ UCV , if and only if

zf 0(z) ∈ UST failed. In 1992, Ronning [110] (independently Ma and Minda [46, 47]) gave

a more applicable one variable characterization of these functions when he was trying to

build up the above bridge between the classes UCV and UST. He showed that

f(z) ∈ UCV, if and only if, for every z ∈ E,

Re

½1 +

zf 00(z)

f 0(z)

¾>

¯zf 00(z)

f 0(z)

¯. (1.4.15)

He defined the following class denoted by ST of analytic functions as follows

ST = g(z) ∈ S∗ : g(z) = zf 0(z), f(z) ∈ UCV .

Ronning [110] conjectured that ST ⊂ UST . Later he proved that neither ST ⊆ UST

nor UST ⊆ ST (see [107, 108]).

In another paper he [109] extended the class UCV by introducing the parameter ρ

(−1 ≤ ρ ≤ 1) and denoted it by UCVρ, consisting of functions f(z): f(z) ∈ A and

23

satisfying

Re

½1 +

zf 00(z)

f 0(z)

¾>

¯zf 00(z)

f 0(z)

¯+ ρ.

Geometrically, we can say that UCVρ is the family of functions f(z) for which the

functional³1 + zf 00(z)

f 0(z)

´takes all the values that lies inside the parabolic region Ω =

w : Re(w − ρ) > |w − 1| , which is symmetric about the real axis and whose vertex is1+ρ2. The function

hρ(z) = 1 +2(1− ρ)

π2

µlog

1 +√z

1−√z

¶2, (1.4.16)

(the branch of√z is chosen such that Im

√z ≥ 0) maps E onto this parabolic region.

Ronning [108], also introduced the class of STρ consisting of functions zf 0(z) for which

f(z) ∈ UCVρ. Note that for ρ = 0, the classes UCVρ and STρ reduce to the main classes

UCV and ST.

In 1999, Kanas and Wisniowska [34, 35], introduced the concepts of κ−uniformly

convexity and κ−starlikeness. They, in fact, imposed a bound κ (a constant κ ≥ 0) on

|ζ| (ζ in the definition of uniformly convex functions as above). They denoted the class

of κ−uniformly convex functions by κ− UCV and defined as:

Let f(z) ∈ A and κ ≥ 0. Then f(z) is in the class κ−UCV , if the image of every circular

arc Υ contained in E, with center ζ, where |ζ| ≤ κ, is convex.

They derived the following one variable characterization of the family κ − UCV as

follows:

Theorem 1.4.7

Let f(z) ∈ A. Then f(z) ∈ κ− UCV, if and only if

Re

½1 +

zf 00(z)

f 0(z)

¾> κ

¯zf 00(z)

f 0(z)

¯, κ ≥ 0, z ∈ E. ¤ (1.4.17)

24

Following the notations of Kanas and Wisniowska [35], we denote by Ωκ the following

family of conic regions

Ωκ =nw = u+ iv : u > κ

p(u− 1)2 + v2

o, (1.4.18)

symmetric about the real axis, with eccentricity equal to 1κ, when κ 6= 0. All of the curves

have one vertex at ( κκ+1

, 0) and the focus (1, 0). If the curve is elliptic one, then the other

vertex is at ( κκ−1 , 0).

Fig. 1.4.1

The domain Ωκ is elliptic for κ > 1, hyperbolic when 0 < κ < 1, parabolic for κ = 1 and

right half plane when κ = 0, see Fig 1.4.1.

From (1.4.17), we can write that for f(z) ∈ κ− UCV

Re

½1 +

zf 00(z)

f 0(z)

¾>

κ

κ+ 1.

25

In another paper Kanas and Wisniowska [34] defined the class κ− ST as follows.

Let f(z) ∈ A. Then f(z) is in the class κ− ST if and only if

Rezf 0(z)

f(z)> κ

¯zf 0(z)

f(z)− 1¯, z ∈ E.

It is important to note that for κ = 0, κ−ST reduces to well-known class S∗ of starlike

functions and for κ = 1, it becomes the class ST discussed earlier. Recent work on these

classes can be found in [4, 21, 56, 103, 118].

1.5 The class of bounded boundary rotation and re-

lated topics

In this section, we study some more concepts in geometric function theory such as func-

tions with bounded boundary and bounded radius rotation and their generalizations.

1.5.1 Functions with bounded boundary rotation

For a simple closed domain with smooth boundary, the boundary rotation σ is defined

as the total variation of the direction angle of the tangent to the boundary curve under

a complete circuit. Thus σ ≥ 2π, with equality, if and only if, the domain is convex.

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. For each real number k ≥ 2, the class

Vk consists of all functions f(z) of the form (1.2.1) and of bounded boundary rotation.

An analytic representation for functions f(z) in the class Vk is given by

2πZ0

¯Re

µ1 +

zf 00(z)

f 0(z)

¶¯dθ ≤ kπ, k ≥ 2.

The classes Vk expand with the increase in k. Paatero [89] showed that for 2 ≤ k ≤ 4, the

functions in the class Vk are univalent, that is, Vk ⊂ S (2 ≤ k ≤ 4). Renyi [104] proved

26

the famous Bieberbach conjecture |an| ≤ n for the class V4. Much later, Pinchuk [93]

and Brannan [9] observed that V4 is properly contained in the class K of close-to-convex

functions. However, for k > 4 each class Vk contains non-univalent functions. It was

shown by Kirwan [37] that the radius of univalence of Vk for k > 4 is tan πk.

For arbitrary k ≥ 2, Lowner [45] obtained the sharp distortion result

(1− r)k2−1

(1 + r)k2+1≤ |f 0(z)| ≤ (1 + r)

k2−1

(1− r)k2+1, |z| = r < 1,

for all f(z) ∈ Vk, with equality sign only for certain rotations of the the wedge mapping

Fk(z) =1

k

"µ1 + z

1− z

¶k2

− 1#. (1.5.1)

This function plays the role of the Koebe function for the class Vk. In particular, F4(z)

is the Koebe function and F2(z) is the half plane mapping l(z) defined by (1.4.8), the

typical extremal function for problems involving convex functions, see [19].

The radius rc of convexity for f(z) ∈ Vk that maps the disk onto a convex domain

was first obtained by Paatero in [90], is given as

rc =k −√k2 − 42

,

and this is best possible as can be seen from the function Fk(z) ∈ Vk defined by (1.5.1).

1.5.2 Functions with bounded radius rotation

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

rotation, if its range has bounded radius rotation(the rotation of the radial vector along

the boundary curve is bounded by k). For each real number k ≥ 2, Rk denotes the class

which consists of all functions f(z) of the form (1.2.1) and of bounded radius rotation.

27

An analytic representation for functions f(z) in the class Rk is given by

2πZ0

¯Re

zf 0(z)

f(z)

¯dθ ≤ kπ, k ≥ 2. (1.5.2)

This class was introduced by Tammi [121] in 1952 and later, it was also studied in

[72, 73, 102].

It is clear that

f(z) ∈ Vk ⇔ zf 0(z) ∈ Rk. (1.5.3)

We observe that R2 ≡ S∗, the class of starlike functions with respect to origin. Also we

can write

f(z) ∈ Rk ⇔zf 0(z)

f(z)∈ Pk, z ∈ E. (1.5.4)

Using the order terminology, Padmanabhan and Parvatham [91] introduced the classes

Vk(ρ) and Rk (ρ) of bounded boundary and bounded radius rotations of order ρ,

0 ≤ ρ < 1, as:

f(z) ∈ Vk(ρ)⇐⇒µ1 +

zf 00(z)

f 0(z)

¶∈ Pk(ρ), z ∈ E, (1.5.5)

and

f(z) ∈ Rk(ρ)⇐⇒zf 0(z)

f(z)∈ Pk(ρ), z ∈ E. (1.5.6)

For ρ = 0, these classes reduce to the parent classes Vk and Rk of functions with bounded

boundary and bounded radius rotations given above. Also the relation (1.5.3) holds

between Vk(ρ) and Rk(ρ).

In our recent published work [86], coauthored with Arif and Mustafa, the following

lemma is proved. This result determines the relationship between the class Vk(ρ) and the

class S∗(ρ).

28

Lemma 1.5.1

Let f(z) ∈ Vk(ρ). Then there exist s1(z), s2(z) ∈ S∗(ρ) such that

f 0(z) =

³s1(z)z

´k4+ 12

³s2(z)z

´k4− 12

, z ∈ E.

Proof. It can easily be shown that f(z) ∈ Vk(ρ), if and only if, there exists f1(z) ∈ Vk

such that

f 0(z) = (f 01(z))1−ρ, z ∈ E, see [91]. (1.5.7)

From representation form due to Brannan [9] , we have

f 01(z) =

³g1(z)z

´k4+12

³g2(z)z

´k4−12

, gi(z) ∈ S∗, i = 1, 2. (1.5.8)

Now, it is shown in [94] that for si(z) ∈ S∗ (ρ), we can write

si (z) = z

∙gi (z)

z

¸1−ρ, gi(z) ∈ S∗, i = 1, 2. (1.5.9)

Using (1.5.8) together with (1.5.9) in (1.5.7), we obtain the required result.

1.5.3 Some related classes with bounded boundary rotation

Using the class Vk, Noor [63, 64] introduced the classes which generalized the concept of

close-to-convex functions and have simple geometrical meanings. We denote these classes

by Nk and Nkk respectively.

1. Let f(z) ∈ A. Then f(z) is said to be in the class Nk, if there is a function g(z) ∈ Vk

such that

Re

½f 0(z)

g0(z)

¾> 0, z ∈ E.

29

Using the same method as that of Kaplan [36], she showed that f(z) ∈ Nk , if and only

ifθ2Z

θ1

Re

½(zf 0(z))0

f 0(z)

¾dθ > −k

2π, z = reiθ, 0 ≤ θ1 < θ2 ≤ 2π. (1.5.10)

Geometrically, it means that the image domain is bounded by a curve Cr and the outward

drawn normal has an angle arg[eiθf 0(reiθ)] at a point on Cr. Then from (1.5.10), it follows

that the angle of the outward drawn normal turns back at most k2π. This is a necessary

condition for a function f(z) to be in the class Nk, see for details [64].

2. Let f(z) ∈ A. Then f(z) is said to be in the class Nkk if there is a function g(z) ∈ Vk

such thatf 0(z)

g0(z)∈ Pk, z ∈ E.

Noor [63] introduced and widely studied this class. We note that , N22 = K, the class of

close-to-convex functions.

Now we switch our discussion to most important concepts of differential subordination

and convolution (Hadamard product). These are the techniques recently used for solving

different problems in analytic function theory. We refer for differential subordination the

book by Miller and Mocanu [50] and for convolution the book by Ruscheweyh [111].

1.6 Differential subordination

In very simple terms, a differential subordination in the complex plane is the general-

ization of a differential inequality on the real line. Obtaining information about the

properties of a function from its derivatives plays an important role in functions of a real

variable. In the field of real-valued functions there are many important theorems dealing

with the theory of differential inequalities. The growth in the field of differential inequal-

ities is a development of the last fifty years. In the theory of complex-valued functions

there have many differential implications in which a characterization of the function is

determined by the differential condition. A simple example is the Noshiro-Warschawski

30

result discussed above in section 1.3. However, until the recent development of the the-

ory of differential subordinations, there has been a scattering of differential implications,

similar to

Re

∙h(z) + α

zh0(z)

h(z)

¸> 0 =⇒ Reh(z) > 0, z ∈ E, (1.6.1)

where h(z) is analytic in E. In 1981, such implications were tackled in the article [52] by

Miller and Mocanu. Since then more than 300 articles on this topic have appeared.

Now we introduce some basic notations, definitions and lemmas which are needed in

our later investigations.

Subordination

Let f(z) and g(z) be in class A. The function f(z) is said to be subordinate to g(z),

written f(z) ≺ g(z), if there exists a function w(z) analytic in E, with w(0) = 0 and

|w(z)| < 1, and such that f(z) = g(w(z)). If g(z) is univalent, then f(z) ≺ g(z), if and

only if f(0) = g(0) and f(E) ⊂ g(E), see [100].

Dominant and best dominant

The univalent function q(z) is called a dominant of the solutions of the differential subor-

dination, if h(z) ≺ q(z) for all solutions h(z) satisfying the given differential subordina-

ton. A dominant eq(z) that satisfies q(z) ≺ eq(z) for all dominants q(z) of the differentialsubordination is called the best dominant.

The implications (1.6.1) and similar results can be solved by using the following

lemmas. These results are very useful tools in our later chapters.

Lemma 1.6.1 [49]

Let u = u1 + iu2 and v = v1 + iv2 and let Ψ (u, v) : D ⊂ C2 → C be a complex-valued

function satisfying the conditions:

(i) Ψ (u, v) is continuous in a domain D ⊂ C2,

31

(ii) (1, 0) ∈ D and Ψ (1, 0) > 0.

(iii) ReΨ (iu2, v1) ≤ 0 whenever (iu2, v1) ∈ D and v1 ≤ −12 (1 + u22) .

If h (z) = 1+c1z+c2z2+· · · is a function that is analytic in E such that (h(z), zh0(z)) ∈ D

and ReΨ (h(z), zh0(z)) > 0 for z ∈ E, then Reh(z) > 0. ¤

Lemma 1.6.2 [51]

Let q(z) be convex in E and Reaq(z) + b > 0, where a, b ∈ C− 0, z ∈ E. If h(z)

is analytic in E with q(0) = h(0) and

h(z) +z h0(z)

ah(z) + b≺ q(z), z ∈ E,

then h(z) ≺ q(z). ¤

Lemma 1.6.3 [51]

Let q(z) be convex in E and j : E 7−→ C with Re j(z) > 0, z ∈ E. If h(z) is analytic in

E and

h(z) + j(z)zh0(z) ≺ q(z), z ∈ E,

then h(z) ≺ q(z). ¤

Lemma 1.6.4 (Rogosinski Lemma [106])

Let f(z) be subordinate to g(z), with

f(z) = 1 +∞Xn=1

anzn and g(z) = 1 +

∞Xn=1

bnzn.

If g(z) is univalent in E and g(E) is convex, then |an| ≤ |b1|. ¤The following lemma is the reverse case of implication (1.6.1).

32

Lemma 1.6.5 [114]

Let h(z) be an analytic function in E with h(0) = 1 and Reh(z) > 0, z ∈ E. Then for

s > 0 and µ 6= −1 (complex),

Re∙h(z) +

szh0(z)

h(z) + µ

¸> 0, for |z| < r0,

where r0 is given by

r0 =|µ+ 1|q

t+ (t2 − |µ2 − 1|2) 12,

t = 2(s+ 1)2 + |µ|2 − 1,

and this result is sharp. ¤By using the principle of subordination, Janowski [33] introduced the class P [A,B]

defined as:

Let h(z) be analytic in E with h(0) = 1. Then h(z) ∈ P [A,B] , if and only if

h(z) ≺ 1 +Az

1 +Bz, z ∈ E,

where −1 ≤ B < A ≤ 1. Note that P [−1, 1] = P, P [−1, 1− 2ρ] = P (ρ). Extending this

idea, Noor [77] discussed the class Pk [A,B] of functions h(z) analytic in E with h(0) = 1

such that

h(z) =

µk

4+1

2

¶h1(z) +

µk

4− 12

¶h2(z), k ≥ 2, (1.6.2)

where hi(z) ∈ P [A,B], i = 1, 2. For k = 2, it reduces to the parent class P [A,B] . For

convenience we take A ∈ C, B ∈ [−1, 0] and A 6= B.

The following lemma will be helpful in our later investigations.

33

Lemma 1.6.6[50]

Let a, b, A ∈ C and B ∈ [−1, 0] satisfy either

Re£a (1 +AB) + b

¡1 +B2

¢¤≥¯aA+ aB +B(b+ b)

¯, (1.6.3)

when B ∈ (−1, 0], or

Re a (1 +A) > 0 and Re [a(1−A) + 2b] ≥ 0, (1.6.4)

when B = −1. If h(z) analytic in E, with h(0) = 1 satisfies

h(z) +zh0(z)

ah(z) + b≺ 1 +Az

1 +Bz, (z ∈ E),

then

h(z) ≺ q(z) ≺ 1 +Az

1 +Bz, (1.6.5)

where

q(z) =1

g(z)− b

a,

is the univalent solution of the differential equation

q(z) +zq0(z)

aq(z) + b=1 +Az

1 +Bz, (z ∈ E)

and

g(z) =

⎧⎪⎪⎨⎪⎪⎩1R0

£1+Bτz1+Bz

¤a(AB−1)

τa+b−1dτ, if B 6= 0,1R0

eaA(τ−1)zτa+b−1dτ, if B = 0.

(1.6.6)

The univalent solution q(z) is the best dominant of the differential subordination (1.6.5).

¤

34

1.6.1 Hypergeometric functions

Prior to proof of Bieberbach conjecture by de Branges [8] there had been only a few

articles in the literature dealing with the relationships between special functions and

univalent function theory. In 1961 Merkes and Scott [48] investigated the starlikeness of

hypergeometric functions, while in 1984 Carlson and Shaffer [13] defined a convolution

operator involving incomplete beta function and obtained results for starlike and related

functions.

Here we present two groups of concepts, the first relating to confluent hypergeometric

functions and the second relating to the Gaussian hypergeometric functions, for more

details see [50].

1. Confluent hypergeometric functions

Let a and c be complex numbers with c 6= −0,−1,−2, . . . , and consider the function

defined by

φ(a, c; z) = 1F1(a, c; z) = 1 +a

c

z

1!+

a(a+ 1)

c(c+ 1)

z2

2!+

a(a+ 1)(a+ 2)

c(c+ 1)(c+ 2)

z3

3!+ . . . . (1.6.7)

This function, called the confluent (or Kummer) hypergeometric function, is analytic in

C and satisfies Kummer’s differential equation

zw00(z) + (c− z)w0(z)− aw(z) = 0.

If we let

(x)n =Γ(x+ n)

Γ(x)=

⎧⎨⎩ x(x+ 1)(x+ 2) . . . (x+ n− 1), n ∈ N,

1, n = 0,

where Γ is the Gamma function, then (1.6.7) can be written in the form

Φ(a, c; z) =∞Xn=0

(a)n(c)n

zn

n!.

35

If Re c > Re a > 0, then

Φ(a, c; z) =Γ(c)

Γ(a)Γ(c− a)

1Z0

τa−1(τ − 1)c−a−1eτzdτ =1Z0

eτzdµ(τ),

where

dµ(τ) =Γ(c)τa−1(τ − 1)c−a−1

Γ(a)Γ(c− a)dτ,

is a probability measure on [0, 1] . In fact

1Z0

dµ(τ) = 1.

2. Gaussian hypergeometric function

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) =∞Xn=0

(a)n(b)n(c)n

zn

n!

called the Gaussian hypergeometric function, is analytic in E and satisfies the Gauss

hypergeometric differential equation

z(1− z)w00(z) + (c− (a+ b+ 1)z)w0(z)− abw(z) = 0.

We list some of the elementary properties of such functions that can be found in [50].

F (a, b, c; z) = F (b, a, c; z), (1.6.8)

cF 0(a, b, c; z) = abF (a+ 1, b+ 1, c+ 1; z), (1.6.9)

F (a, b, c; z) = (1− z)c−a−bF (c− a, c− b, c; z), (1.6.10)

F (a, b, c; z) = (1− z)−aF (a, c− b, c;z

z − 1), (1.6.11)

F (a, b, b; z) = (1− z)−a. (1.6.12)

36

If Re c > Re a > 0, then there is a probability measure on [0, 1] given by

dµ(τ) =Γ(c)τ b−1(τ − 1)c−b−1

Γ(b)Γ(c− b)dτ,

such that

F (a, b, c; z) =

1Z0

(1− τz)−adµ(τ). (1.6.13)

The following lemma due to Bulboaca [11] will be helpful later on.

Lemma 1.6.7[11]

Let c > −1 and ρ0 = maxm−cm+1, 2m−c2(m+1)

≤ ρ ≤ 2m−c+12(m+1)

, then

ρ ≤ bρ(m, c, ρ) =1

m+ 1

∙c+ 1

F (1, 2(m+ 1)(1− ρ), c+ 2, 12)− c+m

¸, (1.6.14)

where F denotes Gauss hypergeometric function. If

½h(z) +

zh0(z)

(m+ 1)h(z) + (c−m)

¾∈ P (ρ),

then h ∈ P (bρ), where bρ is given by (1.6.14). This result is sharp. ¤1.7 Convolution (Hadamard product) and certain lin-

ear operators

The convolution or Hadamard product, of two analytic functions f(z) = z +∞Pn=2

an zn

and g(z) = z +∞Pn=2

bn zn is the function (f ∗ g)(z) with the series representation of the

form

(f ∗ g)(z) = f(z) ∗ g(z) = z +∞Xn=2

an bn zn, |z| < 1.

37

The convolution obeys the algebraic properties of ordinary multiplication. Convoluting

the power series of l(z) = z+∞Pn=2

zn = z1−z with that of f(z) = z+

∞Pn=2

an zn, we see that

f(z) ∗ l(z) = f(z),

for all f(z) ∈ A. It means that the mapping l(z) behaves an identity mapping for convolu-

tion or Hadamard product. The convolution or Hadamard product is such an important

tool in geometric function theory that so many complicated problems are solved very

easily by using it.

Convolution has a very interesting history in geometric function theory. It began

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

Polya and Schoenberg [95] in 1958. In 1973, Ruscheweyh and Sheil-Small [113] settled

this conjecture, in a very complicated paper. It leads to a wealth of results, including

both the Polya-Schoenberg conjecture and its analogue for the space of close-to-convex

functions. These results are immediate consequences of another generalized result, which

takes a central role in the theory of convolution and is given as below.

Lemma 1.7.1 [111]

(i) Let f(z) ∈ C and g(z) ∈ S∗. Then, for any function F (z) analytic in E with F (0) = 1,

we havef(z) ∗ g(z)F (z)f(z) ∗ g(z) ∈ co (F (E)) , z ∈ E,

(ii) Let f(z) and g(z) be starlike of order1

2. Then, for any function F (z) analytic in E

with F (0) = 1, we have

f(z) ∗ g(z)F (z)f(z) ∗ g(z) ∈ co (F (E)) , z ∈ E,

( co stands for the closed convex hull of a set). ¤

38

Lemma 1.7.2[115]

If h(z) is analytic in E, h(0) = 1 and Reh(z) > 12, z ∈ E, then for any function F (z),

analytic in E with F (0) = 1, the function (h ∗ F )(z) ∈ co (F (E)), z ∈ E.

1.7.1 Certain linear operators defined in terms of convolution

The study of operators can be traced back to 1916 provided by Alexander, defined by

(1.4.5). Later, Libera in 1965 discussed another integral operator and studied its effects on

various classes of univalent functions. Bernardi generalized this operator and investigated

its interesting aspects. A. E. Livingston observed the converse case of Libera’s operator.

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

study of operators, plays an important role in geometric function theory. A wealth of

literature on operators is now available and it can easily occupy space of a book. A large

number of classes of analytic functions are defined by means of different 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) ∈ A, we consider the integral operator

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

zc

zZ0

tc−1f (t) dt, c > −1, (see [28]). (1.7.1)

The operator Lc, when c ∈ N was introduced by Bernardi [7]. In particular, the operator

L1 was studied earlier by Libera [40] and Livingston [44].

39

Ruscheweyh derivative

Let f (z) ∈ A. Denote by Dm : A→ A the operator defined by

Dmf (z) =z

(1− z)m+1∗ f (z) , for m ∈ N0 = N ∪ 0, (1.7.2)

= z +∞Xn=2

(m+ 1)n−1(n− 1)! an zn.

It is obvious that D0f (z) = f (z) , D1f (z) = zf 0 (z) and

Dmf (z) =z (zm−1f (z))

(m)

m!, for all m ∈ N0.

The following identity can easily be settled

(m+ 1)Dm+1f (z) = mDmf (z) + z (Dmf (z))0 . (1.7.3)

The operator Dm : A −→ A was originally introduced by Ruscheweyh [112] and named

as mth-order Ruscheweyh derivative by Al-Amiri [2].

Noor integral operator

Analogous to Ruscheweyh derivative of order m, Noor [71], Noor and Noor [82], defined

and studied an integral operator Im : A→ A, as follows.

Let fm(z) = z(1−z)m+1 , (m ∈ N0), and f

(−1)m (z) be defined such that

fm(z) ∗ f (−1)m (z) =z

(1− z)2.

Then

Imf(z) = f(z) ∗ f (−1)m (z) =

∙z

(1− z)m+1

¸(−1)∗ f(z).

40

We note that I0f(z) = zf 0(z), I1f(z) = f(z). The operator Im was introduced by Noor

[71] and Liu [42] named it "Noor integral operator" of f(z) of orderm, see also [15, 42, 81].

This operator has attracted the attention of many renowned mathematician from different

parts of the world and it is a very useful tool in defining several classes of analytic

functions.

Carlson and Shaffer operator

Let φ(a, c; z) = z +∞Pn=2

(a)n(c)n

zn be the incomplete beta function. Carlson and Shaffer [13]

introduced a linear operator L(a, c) : A −→ A by

L(a, c)f(z) = φ(a, c; z) ∗ f(z). (1.7.4)

From (1.7.4) it can be easily verified that L (2, 1) f (z) = zf0(z) and

z (L(a, c)f(z))0= aL(a+ 1, c)f(z)− (a− 1)L(a, c)f(z).

We note that L(2,m + 1)f(z) = Imf(z) and L(m + 1, 1)f(z) = Dmf(z). For detail we

refer [75].

Hohlov operator

Hohlov [31] introduced the convolution operator

Ha,b,c(f)(z) = 2F1(a, b; c; z) ∗ f(z). (1.7.5)

This operator contains most of the known linear integral or differential operators as

special cases. In particular, if a = 1 in (1.7.5), then H1,b,c reduces to that of Carlson and

Shaffer [13].

41

Fractional calculus operator

The fractional derivative of order α is defined [87] for a function f(z) by

Dαz f(z) =

1

Γ (1− α)

d

dz

Z z

0

f(ξ)

(z − ξ)αdξ, 0 ≤ α < 1,

where f(z) is analytic function in a simply connected domain of the z- plane containing

the origin and the multiplicity of (z − ξ)−α is removed by requiring log(z − ξ) to be real

when (z − ξ) > 0. Using Dαz f(z), Owa et.al [88], introduced the operator Lα : A 7−→ A,

known as the extension of fractional derivative and fractional integral operator, as follows

Lαf(z) = Γ(2− α) zαDαz f(z), 0 ≤ α < 1

= z +∞Xn=2

Γ(n+ 1) Γ(2− α)

Γ(n+ 1− α)an zn (1.7.6)

= φ(2, 2− α; z) ∗ f(z), α 6= 2, 3, . . . , (1.7.7)

where L0f(z) = f(z), see also [116] and [117]. We can also write Dαλf(z), λ ≥ 0, in terms

of Lαf(z) by using (1.7.6) as:

Dαλf(z) = (1− λ)Lαf(z) + λz (Lαf(z))

0

= z +∞Xn=2

Γ(n+ 1) Γ(2− α)

Γ(n+ 1− α)(1 + λ(n− 1))an zn, z ∈ E. (1.7.8)

From (1.7.7) and (1.7.8), Dαλf(z) can be written as

Dαλf(z) = φ(2, 2− α; z) ∗Ψλ(z) ∗ f(z), (1.7.9)

42

where

Ψλ(z) =z − (1− λ) z2

(1− z)2.

Note : All the references for definitions, theorems and lemmas are given and if there is

any missing it can be seen in [19, 28, 100]. It is also important to note that nothing is

produced by author himself in the first chapter.

43

Chapter 2

On classes of κ-uniformly

close-to-convex and related functions

This chapter is devoted to the class of κ-uniformly close-to-convex, quasi-convex func-

tions and some of their generalizations in various aspects. The idea of uniformly close-to-

convex functions was first provided by Kumar and Ramesha [38] in 1994. They discussed

the subordination properties of uniformly convex and uniformly close-to-convex func-

tions. Later, Suberamanian et.al. [120] extended the idea of uniformly close-to-convex

functions by introducing a parameter ρ (−1 ≤ ρ < 1). They also discussed the nat-

ural analogue of uniformly close-to-convex functions known as uniformly quasi-convex

function. They gave the geometrical interpretations of these functions and showed that

uniformly close-to-convex functions of order ρ contained the usual classes of UCVρ and

STρ of uniformly convex and parabolic starlike functions of order ρ as discussed before

in chapter 1. They also provided some sufficient conditions for functions to be in the

classes of uniformly close-to-convex functions of order ρ. Some convolution properties

along with coefficient estimates of uniformly close-to-convex and uniformly quasi-convex

functions were also be the points of their investigations. Recently in [85], we extend the

idea of κ−uniformly close-to-convexity by introducing the concepts of fractional calculus

operator and complex order. The concept of fractional integral and fractional derivative

44

was introduced in geometric function theory by Owa [87]. Later, Owa and Srivastava

[88], introduced the fractional calculus operator denoted by Lα.

In this chapter, we introduce and study the classes κ−UKη(λ, α) and κ−UQη(λ, α)

of κ-uniformly close-to-convex and κ-uniformly quasi-convex functions of complex or-

der. Several inclusion results, convolution properties, coefficient estimates and sufficient

conditions for these classes are the main topics of our investigations in this chapter. We

observe that several well-known results are special cases of our results.

2.1 The class of κ-uniformly close-to-convex func-

tions of complex order

This section comprises the results concerning the class κ−UKη(λ, α) of κ-uniformly close-

to-convex functions of complex order. We study some inclusion relationships, convolution

properties, sufficiency condition and coefficient estimates of this class.

Throughout this chapter we assume that κ ≥ 0, λ ≥ 0, 0 ≤ α < 1 and η ∈ C− 0 unless

otherwise mentioned.

Now, we define the following classes involving the operator Dαλ .

Definition 2.1.1

Let f(z) ∈ A. Then f(z) ∈ UCVη(λ, α), if and only if

Reµ1 +

1

η

µz (Dα

λf(z))00

(Dαλf(z))

0

¶¶> κ

¯1

η

µz (Dα

λf(z))00

(Dαλf(z))

0

¶¯, z ∈ E. (2.1.1)

Definition 2.1.2

Let f(z) ∈ A. Then f(z) ∈ κ− STη(λ, α), if and only if

Reµ1 +

1

η

µz (Dα

λf(z))0

Dαλf(z)

− 1¶¶

> κ

¯1

η

µz (Dα

λf(z))0

Dαλf(z)

− 1¶¯, z ∈ E. (2.1.2)

45

Definition 2.1.3

Let f(z) ∈ A. Then f(z) ∈ κ−UKη(λ, α), if and only if there exists g(z) ∈ κ−ST1(λ, α)

Reµ1 +

1

η

µz (Dα

λf(z))0

Dαλg(z)

− 1¶¶

> κ

¯1

η

µz (Dα

λf(z))0

Dαλg(z)

− 1¶¯, z ∈ E. (2.1.3)

It is clear from (2.1.1) and (2.1.2) that

f(z) ∈ κ− UCVη(λ, α)⇔ zf 0(z) ∈ κ− STη(λ, α). (2.1.4)

Various known classes studied in earlier work[1, 6, 34, 35, 123], appear as special cases

of the classes defined above. Some are given as follows

(i) 0− UCVη(0, 0) ≡ Cη ≡ 0− S∗η(1, 0), 0− S∗η(0, 0) ≡ S∗η and 0− UKη(0, 0) ≡ Kη.

(ii) κ − UCV1(0, 0) ≡ κ − UCV ≡ κ − ST1(1, 0), κ − ST1(0, 0) ≡ κ − ST and 1 −

UK1(0, 0) ≡ UK, the class of uniformly close-to-convex functions as considered by Kumar

and Ramesha [38]. We now discuss for these classes the following.

Geometrical interpretation

A function f(z) ∈ A is in the class κ − UCVη(λ, α) and κ − STη(λ, α), if and only ifz(Dα

λf(z))0 0

(Dαλf(z))

0 ,z(Dα

λf(z))0

Dαλf(z)

andz(Dα

λf(z))0

Dαλg(z)

respectively, take all the values in the conic domain

Ωκ,η with 0 < Re η ≤ κ+ 1 such that

Ωκ,η = ηΩκ + (1− η) , (2.1.5)

where

Ωκ =

½u+ iv : u > κ

q(u− 1)2 + v2

¾.

The domain Ωκ,η is elliptic for κ > 1, hyperbolic when 0 < κ < 1, parabolic for κ = 1

and right half plane when κ = 0.

By using essentially the same method as in [34, 35], we construct the functions which

46

play the role of extremal functions for the conic regions of complex order. For the sake

of completeness, all the details are given. Denoting by hκ,η(z) ∈ P and hκ(z) ∈ P the

functions such that

hκ,η(E) = Ωκ,η and hκ(E) = Ωκ.

It is obvious that

h0,η(z) =1 + (2η − 1) z

1− z, z ∈ E,

and for the case of parabolic region

h1,η(z) = 1 +2η

π2

µlog

1 +√z

1−√z

¶2, z ∈ E.

Now, we shall provide an explicit form of functions which takes E onto the region

Ωκ =

(µ+ iν :

(1− κ2)2

κ2

µµ+

κ2

1− κ2

¶2− (1− κ2)ν2 > 1, µ > 0

), 0 < κ < 1.

(2.1.6)

The transformation

ω0(z) =

µ1 +√z

1−√z

¶ 2πcos−1 κ

,

where the branch of√z is the principle one, maps E onto the angular region of width

cos−1 κ. Also, the map

ω1(z) =1

2

µω0(z) +

1

ω0(z)

¶transforms the angular region onto a domain which is interior to the right branch of the

hyperbola with vertex at the point ω1 = κ. Lastly, the mapping

ωhyper(z) =1

1− κ2ω1(z)−

κ2

1− κ2

47

is only the normalization and maps the above domain onto the interior of the hyperbola,

given by (2.1.6). Thus

hκ(z) = ωhyper(z) =1

2(1− κ2)

"µ1 +√z

1−√z

¶ 2πcos−1 κ

+

µ1−√z1 +√z

¶ 2πcos−1 κ

#− κ2

1− κ2.

This function can be equivalently written as

hκ(z) =1

1− κ2cosh

µ2

πcos−1 κ log

µ1 +√z

1−√z

¶¶− κ2

1− κ2

=1

1− κ2cosh

µ4

πcos−1 κ tanh−1

√z

¶− κ2

1− κ2

= 1 +2

1− κ2

∙cosh2

µ2

πcos−1 κ tanh−1

√z

¶− 1¸

= 1 +2

1− κ2sinh2

µ2

πcos−1 κ tanh−1

√z

¶.

Therefore, from (2.1.5), we can write

hκ,η(z) = 1 +2η

1− κ2sinh2

∙µ2

πarccosκ

¶arctanh

√z

¸, 0 < κ < 1.

Next, we find that for κ > 1,

hκ,η(z) = 1 +η

κ2 − 1 sin

⎛⎜⎜⎝ π

2R(t)

u(z)√tZ

0

1√1− x2

q1− (tx)2

dx

⎞⎟⎟⎠+ η

κ2 − 1 ,

where u(z) = z−√t

1−√tz, t ∈ (0, 1), z ∈ E and z is chosen such that κ = cosh

³πR0(t)4R(t)

´, R(t)

is the Legendre’s complete elliptic integral of the first kind and R0(t) is complementary

integral of R(t)maps E onto the elliptic region, for more details see [34, 35]. Summarizing

48

the above discussion, we can write

hκ,η(z) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1+(2η−1)z1−z , κ = 0,

1 + 2ηπ2

³log 1+

√z

1−√z

´2, κ = 1,

1 + 2η1−κ2 sinh

2£¡

2πarccosκ

¢arctanh

√z¤, 0 < κ < 1,

1 + ηκ2−1 sin

⎛⎝ π2R(t)

u(z)√tR0

1√1−x2√1−(tx)2

dx

⎞⎠+ ηκ2−1 , κ > 1.

(2.1.7)

2.1.1 Some properties of the class κ−STη(λ, α) and κ−UKη(λ, α)

Now we study some basic properties of the class κ − STη(λ, α) which are useful in our

subsequent results for the class κ− UKη(λ, α). Here we will study the inclusion results

and some other interesting properties of these classes by changing various parameters.

The tools of convolution and subordination are used to study these properties.

Theorem 2.1.1

Let Lαf(z) be in the class κ− STη(λ, α), 0 < η ≤ 1. Then f(z) ∈ κ− STη(λ, α). ¤Proof. Using (1.7.7) and (1.7.9), we can write Dα

λf(z) in terms of DαλLαf(z) as

Dαλf(z) = φ(2− α, 2; z) ∗Dα

λLαf(z), (2.1.8)

z (Dαλf(z))

0 = φ(2− α, 2; z) ∗ z (DαλLαf(z))

0 . (2.1.9)

Since

zφ0 (2− α, 2; z) = φ (2− α, 1; z) =z

(1− z)2−α∈ S∗

³α2

´⊆ S∗.

49

This implies that φ(2− α, 2; z) is a convex function. Therefore by using (2.1.8), (2.1.9),

and Lemma 1.7.1(ii), we have,

z (Dαλf(z))

0

Dαλf(z)

=φ(2− α, 2; z) ∗ z (Dα

λLαf(z))0

φ(2− α, 2; z) ∗DαλLαf(z)

=

φ(2− α, 2; z) ∗ z (DαλLαf(z))

0

DαλLαf(z)

DαλLαf(z)

φ(2− α, 2; z) ∗DαλLαf(z)

∈ co

½z (Dα

λLαf(z))0

DαλLαf(z)

(E)

¾⊆ Ωκ,η.

Thus f(z) is in the class κ− STη(λ, α). ¥

Lemma 2.1.1

Let f(z) ∈ κ− STη(λ, α). Then Dαλf(z) ∈ S∗η

¡12

¢for κ ≥ 1, 0 < η ≤ 1. ¤

Proof. Let f(z) ∈ κ− STη(λ, α). Then

Re

(1 +

1

η

Ãz (Dα

λf(z))0

Dαλf(z)

− 1!)

>κ

κ+ 1.

Sine κκ+1≥ 1

2for κ ≥ 1, so we have

Re

(1 +

1

η

Ãz (Dα

λf(z))0

Dαλf(z)

− 1!)

>κ

κ+ 1≥ 12.

Hence,

Dαλf(z) ∈ S∗η

µ1

2

¶.

This completes the proof. ¥Now, we give the coefficient estimates of the functions that are in the class κ−STη(λ, α).

These estimates are essential in order to have the coefficient estimates for the class

κ− UKη(λ, α) which will be provided later on.

50

Theorem 2.1.2

Let f(z) ∈ κ− STη(λ, α) and be given by (1.2.1). Then

|an| ≤Γ(n+ 1− α)

(1 + λ(n− 1))Γ(n+ 1)Γ(2− α)

(|δκ,η|)n−1(n− 1)! , n ≥ 2 ,

where

δκ,η =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

8η(cos−1 κ)2

π2(1− κ2), 0 ≤ κ < 1,

8η

π2, κ = 1,

π2η

4√t(κ2 − 1)R2(t)(1 + t)

, κ > 1.

(2.1.10)

This result is sharp for n = 2 or κ = 0. ¤Proof. Let F (z) = (1−λ)Lαf(z)+λz (Lαf(z))

0. Then, from definition of κ−STη(λ, α),

we havez (F (z))

0

F (z)≺ hκ,η(z), for z ∈ E,

where hκ,η(z) is given by (2.1.7).

Now F (z) can be written as⎧⎪⎪⎪⎨⎪⎪⎪⎩F (z) = z +

P∞n=2Anz

n, z ∈ E,

An =(1+λ(n−1))Γ(n+1)Γ(2−α)

Γ(n+1−α) an, n ≥ 2.

(2.1.11)

Setz (F (z))

0

F (z)= h(z) = 1 +

∞Xn=1

cnzn. (2.1.12)

Then

zF0(z) = h(z)F (z). (2.1.13)

51

Using (2.1.11), (2.1.12) and (2.1.13), we find that

(n− 1)An =n−1Xj=1

cn−jAj ,

where we choose A1 = 1. From Lemma 1.6.4, we have

|cn| ≤ |δκ,η| ,

where δκ,η is given by (2.1.10) . In particular, for n = 2, 3, 4, we obtain

|A2| ≤ |δκ,η| , |A3| ≤|δκ,η| (|δκ,η|+ 1)

2!

and

|A4| ≤|δκ,η| (|δκ,η|+ 1) (|δκ,η|+ 2)

3!

respectively. Using the principle of mathematical induction, we obtain

|An| ≤(|δκ,η|)n−1(n− 1)! .

Now by using (2.1.11), we obtain the required result. ¥

Theorem 2.1.3

Let Lαf(z) be in the class κ− UKη(λ, α). Then f(z) ∈ κ− UKη(λ, α). ¤Proof. Let Lαf(z) ∈ k−UKη(λ, α). Then there exists Lαg(z) in the class κ−ST1(λ, α)

such that

Re

Ã1 +

1

η

Ãz (Dα

λLαf(z))0

DαλLαg(z)

− 1!!

> κ

¯¯1ηÃz (Dα

λLαf(z))0

DαλLαg(z)

− 1!¯¯ . (2.1.14)

As Lαg(z) ∈ κ − ST1(λ, α), then by Theorem 2.1.1, g (z) ∈ κ − ST1(λ, α) and hence

52

Dαλg(z) ∈ S∗. Also by using (2.1.14),

DαλLαg(z) ∈ S∗1(

kk+1) ⊆ S∗. By using (2.1.8), (2.1.9) and Lemma 1.7.1(ii), we have,

z (Dαλf(z))

0

Dαλg(z)

=φ(2− α, 2; z) z (Dα

λLαf(z))0

φ(2− α, 2; z) DαλLαg(z)

=

φ(2− α, 2; z)z (Dα

λLαf(z))0

DαλLαg(z)

DαλLαg(z)

φ(2− α, 2; z) DαλLαg(z)

∈ co

(z (Dα

λLαf(z))0

DαλLαg(z)

(E)

)⊆ Ωκ,η .

Thus f(z) is in the class κ− UKη(λ, α). ¥

Theorem 2.1.4

Let 0 ≤ α1 ≤ α2 < 1. Then, for κ ≥ 1, 0 < η ≤ 1,

κ− UKη(λ, α2) ⊆ κ− UKη(λ, α1).

Proof. Let f(z) ∈ κ− UKη(λ, α2). Then from (1.7.9), we have

Dα1λ f(z) = φ (2− α2, 2− α1; z) ∗Dα2

λ f(z). (2.1.15)

From which, we can have

z (Dα1λ f(z))0 = φ (2− α2, 2− α1; z) ∗ z (Dα2

λ f(z))0 . (2.1.16)

53

Since φ (2− α2, 2− α1; z) ∈ S∗¡12

¢, see [41], we use Lemma 2.1.1 to haveDα2

λ f(z) belongs

to S∗η¡12

¢⊆ S∗

¡12

¢for κ ≥ 1. Now using (2.1.15), (2.1.16) and Lemma 1.7.1(i), we obtain

z (Dα1λ f(z))0

Dα1λ f(z)

=

φ (2− α2, 2− α1; z) ∗z (Dα2

λ f(z))0

Dα2λ f(z)

Dα2λ f(z)

φ (2− α2, 2− α1; z) ∗Dα2λ f(z)

∈ co

½z (Dα2

λ f(z))0

Dα2λ f(z)

(E)

¾⊆ Ωκ,η.

This implies thatz (Dα1

λ f(z))0

Dα1λ f(z)

∈ Ωκ,η and hence we have the desired result. ¥

Theorem 2.1.6

For 0 < Re η ≤ κ+ 1,

κ− UKη(1, α) ⊂ κ− UKη(0, α).

Proof. Let f(z) ∈ κ− UKη(1, α). Then, by¡z (Lαf(z))

0¢0(Lαg(z))

0 ≺ hκ,η(z). (2.1.17)

Taking z(Lαf(z))0

Lαg(z)= h(z), we see that h(z) is analytic and h(0) = 1. Simple computations

together with (2.1.17) shows that

¡z (Lαf(z))

0¢0(Lαg(z))

0 = h(z) +Lαg(z)

z (Lαg(z))0 zh

0(z) ≺ hκ,η(z),

where hκ,η(z) are given in (2.1.7). Since Rez(Lαg(z))

0

Lαg(z)> 0 and hκ,η(z) are convex in E,

then by Lemma 1.6.3, we write

h(z) ≺ hκ,η(z).

Also it can be easily shown that g(z) ∈ κ − STη(1, α) ⊂ κ − STη(0, α). Hence f(z) ∈

κ− UKη(0, α). ¥

54

2.1.2 Convolution invariance with convex function

Convoluting various classes with convex function is an interesting problem in geometric

function theory. In the following theorem, we study the convolution of convex functions

with the functions from the classes κ− UKη(λ, α).

Theorem 2.1.7

Let f(z) ∈ κ− UKη(λ, α), 0 < η ≤ 1, and h(z) ∈ C. Then

f(z) ∗ h(z) ∈ κ− UKη(λ, α).

Proof. Let f(z) ∈ κ− UKη(λ, α). Then, by definition

z (Dαλf(z))

0

Dαλg(z)

∈ Ωκ,η,

where g(z) ∈ κ− ST1(λ, α). To obtain the desired result, it is sufficient to show

z (h(z) ∗Dαλf(z))

0

h(z) ∗Dαλg(z)

∈ Ωκ,η.

Since Dαλg(z) ∈ S∗η ⊆ S∗ and h(z) ∈ C. Then, by using Lemma 1.7.1(i), we have

z (h(z) ∗Dαλf(z))

0

h(z) ∗Dαλg(z)

=

h(z) ∗ z (Dαλf(z))

0

Dαλg(z)

Dαλg(z)

h(z) ∗Dαλg(z)

∈ co

½z (Dα

λf(z))0

Dαλg(z)

(E)

¾⊆ Ωκ,η .

Hence f(z) ∗ h(z) ∈ κ− UKη(λ, α). ¥Now we derive coefficient estimates of functions in the class κ− UKη(λ, α).

55

Theorem 2.1.8

Let f(z) ∈ A be defined by (1.2.1), is in the class κ− UKη(λ, α). Then

|an| ≤Γ(n+ 1− α)

(1 + λ(n− 1))Γ(n+ 1)Γ(2− α)

"(|δκ,1|)n−1

n!+|δκ,η|n

n−1Xj=1

(|δκ,1|)j−1(j − 1)!

#, n ≥ 2,

where δκ,η is given by (2.1.10). This result is sharp for κ = 0 or n = 2. ¤

Proof. Since f(z) ∈ κ− UKη(λ, α), then there exists g(z) = z +∞Xn=2

bnzn belonging to

the class ST1(λ, α), such that

z (F (z))0

G(z)≺ hκ,η(z), for z ∈ E,

where F (z) is given by (2.1.11) and G(z) can be written as⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩G(z) = z +

∞Xn=2

Bnzn, z ∈ E,

Bn =(1 + λ(n− 1))Γ(n+ 1)Γ(2− α)

Γ(n+ 1− α)bn, n ≥ 2.

(2.1.18)

Letz (F (z))

0

G(z)= q(z) = 1 +

∞Xn=1

dnzn, for z ∈ E. (2.1.19)

Then

zF0(z) = q(z)G(z). (2.1.20)

Using (2.1.18), (2.1.19) and (2.1.20) we find that

nAn = Bn +n−1Xj=1

dn−jBj, n ≥ 2,

56

where A1 = 1 and B1 = 1. This implies that

n |An| ≤ |Bn|+n−1Xj=1

|dn−j| |Bj| , n ≥ 2. (2.1.21)

From Lemma 1.6.4, together with Theorem 2.1.2, we have

|dj| ≤ |δκ,η| , j ≥ 1,

and

|Bn| =(1 + λ(n− 1))Γ(n+ 1)Γ(2− α)

Γ(n+ 1− α)|bn| , n ≥ 2

≤ (|δκ,1|)n−1(n− 1)! , n ≥ 2.

Therefore (2.1.21) becomes

|An| ≤(|δκ,1|)n−1

n!+|δκ,η|n

n−1Xj=1

(|δκ,1|)j−1(j − 1)! ,

and hence from (2.1.11), we obtain the required result

|an| ≤Γ(n+ 1− α)

(1 + λ(n− 1))Γ(n+ 1)Γ(2− α)

"(|δκ,1|)n−1

n!+|δκ,η|n

n−1Xj=1

(|δκ,1|)j−1(j − 1)!

#, n ≥ 2.¥

1. For α = 0, λ = 0, and κ = 0, in the above theorem, we have the known coefficient

estimates proved in [3].

2. For α = 0, λ = 0, κ = 1 and η = 1, in the above theorem, we obtain the following.

57

Corollary 2.1.1

Let f(z) ∈ A be defined by (1.2.1), is in the class UK. Then

|an| ≤( 8π2)n−1

n!+

8

nπ2

n−1Xj=1

( 8π2)j−1

(j − 1)! , n ≥ 2.

The estimates are sharp for n = 2. ¤

2.1.3 Sufficient condition for functions in κ− UKη(λ, α)

Theorem 2.1.9

Let f(z) ∈ A and be of the form (1.2.1), satisfying

∞Xn=2

n |An| ≤|η|2,

where An are given by (2.1.11). Then f(z) ∈ κ− UKη(λ, α), for 0 ≤ κ ≤ 1. ¤Proof. Setting g(z) = z, we have

z(Dαλf(z))

0

Dαλg(z)

= (Dαλf(z))

0 = 1 +∞Xn=2

nAnzn−1,

so that for z in E,

κ

¯1

η

µz(Dα

λf(z))0

Dαλg(z)

− 1¶¯≤ 1

|η| |(Dαλf(z))

0 − 1|

≤ 1

|η|

∞Xn=2

n |An| ≤1

|η|

Ã|η|−

∞Xn=2

n |An|!

≤ Re

µ1 +

1

η((Dα

λf(z))0 − 1)

¶.

58

Thus z(Dαλf(z))

0

Dαλg(z)

∈ Ωκ,η and hence f(z) ∈ κ− UKη(λ, α). ¥Choosing α = 0, λ = 0, η = 1− ρ, in the above result, we have the sufficient condition

proved in [38].

2.2 The class κ− UQCη(λ, α)

In this section,we deal with class κ − UQCη(λ, α) of uniformly quasi-convex functions

which is the natural analogue of the class κ − UKη(λ, α) in the sense of Alexander’s

result, see [59]. These functions play a central role in geometric function theory and pro-

vide a natural generalization of convex functions. We prove inclusion results, coefficient

estimates and sufficiency condition for this class.

We define this class in the same spirit as that of Noor [59]. It is defined as

κ− UQCη(λ, α) = f(z) ∈ A : zf 0(z) ∈ κ− UKη(λ, α), z ∈ E . (2.2.1)

2.2.1 Inclusion results

Using the properties of the class κ− UKη(λ, α) as discussed in the previous section, we

study the following properties of the class κ− UQCη(λ, α).

Theorem 2.2.1

Let Lαf(z) be in the class κ− UQCη(λ, α), 0 < η ≤ 1. Then f(z) ∈ κ− UQCη(λ, α). ¤Proof. By using (2.2.1) and Theorem 2.1.3, we have

Lαf(z) ∈ κ− UQCη(λ, α)

⇔ z (Lαf(z))0 ∈ κ− UKη(λ, α)

⇔ Lα(zf0(z)) ∈ κ− UKη(λ, α)

⇒ zf0(z) ∈ κ− UKη(λ, α)

⇔ f(z) ∈ κ− UQCη(λ, α).

59

Thus f(z) ∈ κ− UQCη(λ, α). ¥

Theorem 2.2.2

Let 0 ≤ α1 ≤ α2 < 1. Then, for κ ≥ 1, 0 < η ≤ 1,

κ− UQCη(λ, α2) ⊆ κ− UQCη(λ, α1).

Proof. Let f(z) ∈ κ−UQCη(λ, α2). Then, by using (2.2.1) and Theorem 2.1.4, we have

f(z) ∈ κ− UQCη(λ, α2)

⇔ zf 0(z) ∈ κ− UKη(λ, α2)

⇒ zf 0(z) ∈ κ− UKη(λ, α1)

⇔ f(z) ∈ κ− UQCη(λ, α1).

Thus f(z) ∈ κ− UQCη(λ, α1) and hence we obtain the required result. ¥Next, we prove that the class κ−UQCη(λ, α) is preserved under the convolution with

convex functions.

Theorem 2.2.3

Let f(z) ∈ κ− UQCη(λ, α), 0 < η ≤ 1, and h(z) ∈ C. Then

f(z) ∗ h(z) ∈ κ− UQCη(λ, α).

Proof. Let f(z) ∈ κ− UQCη(λ, α). Then, by using (2.2.1), we have

zf 0(z) ∈ κ− UKη(λ, α).

60

Since h(z) ∈ C, so by using Theorem 2.1.7, we obtain

zf 0(z) ∗ h(z) ∈ κ− UKη(λ, α)

⇔ z (f(z) ∗ h(z))0 ∈ κ− UKη(λ, α).

Thus, by using (2.2.1), we obtain the required result. ¥

2.2.2 Coefficient Estimates

Now, we discuss the coefficient estimates of functions in the class κ−UQCη(λ, α). Using

the estimates given in Theorem 2.1.8, we have the following.

Theorem 2.2.4

Let f(z) ∈ κ− UQCη(λ, α) and be given by (1.2.1). Then

|an| ≤Γ(n+ 1− α)

n2(1 + λ(n− 1))Γ(n+ 1)Γ(2− α)

"(|δκ,1|)n−1(n− 1)! + |δk,η|

n−1Xj=1

(|δk,1|)j−1(j − 1)!

#, n ≥ 2 .

This result is sharp for n = 2 or κ = 0. ¤

Proof. Let f(z) ∈ κ−UQCη(λ, α). Then from relation (2.2.1), zf 0(z) ∈ κ−UKη(λ, α).

Now using Theorem 2.1.8, we obtain

n |an| ≤Γ(n+ 1− α)

(1 + λ(n− 1))Γ(n+ 1)Γ(2− α)

"(|δk,1|)n−1

n!+|δk,η|n

n−1Xj=1

(|δk,1|)j−1(j − 1)!

#, n ≥ 2,

where δκ,η is given by (2.1.10). Thus we have

|an| ≤Γ(n+ 1− α)

n2(1 + λ(n− 1))Γ(n+ 1)Γ(2− α)

"(|δκ,1|)n−1(n− 1)! + |δk,η|

n−1Xj=1

(|δk,1|)j−1(j − 1)!

#, n ≥ 2,

which is the required coefficient estimates for the class κ− UQCη(λ, α). ¥

61

2.3 Some applications of convolution invariance

Here we will see some interesting applications of convolution invariance property proved

earlier in Theorem 2.1.7 and Theorem 2.2.3.

Theorem 2.3.1

The classes κ−UKη(λ, α) and κ−UQCη(λ, α) are invariant under the following operators.

(i) . f1(z) =

zZ0

f (ξ)

ξdξ.

(ii) . f2(z) =2

z

zZ0

f (ξ) dξ.

(iii) . f3(z) =

zZ0

f (ξ)− f (xξ)

ξ − xξdξ, |x| ≤ 1, x 6= 1.

(iv) . f4(z) =1 + c

zc

zZ0

ξc−1f (ξ) dξ, Re c > 0.

Proof. We write fi(z) = f(z)∗ψi (z), where ψi (z) , 1 ≤ i ≤ 4, are convex and given by

ψ1(z) = − log(1− z),

ψ2(z) =−2 [z − log(1− z)]

z,

ψ3(z) =1

1− xlog

µ1− xz

1− z

¶, |x| ≤ 1, x 6= 1,

ψ4(z) =∞Xn=1

1 + c

n+ czn, Re c > 0.

The results are followed by applying Theorem 2.1.7 and Theorem 2.2.3. ¥

62

Chapter 3

Some properties of a subclass of

analytic functions

Functions with bounded turning, that is, the functions with derivative having real part

greater than zero, and their generalizations have very close connection to various classes of

analytic univalent functions. These classes have been considered bymanymathematicians

such as Noshiro [57]and Warchawski [122], Chichra [14], Goodman [28] and Noor [84].

In this chapter, we define and discuss a certain subclass of analytic functions related

with the functions with bounded turning. An inclusion result, a radius problem, invari-

ance under certain integral operators and some other interesting properties for this class

will be discussed. We define the class ePk(β) as follows.

Definition 3.1.1

Let h(z) be analytic in E with h(0) = 1. Then h(z) ∈ ePk(β), k ≥ 2, 0 < β ≤ 1, if and

only if, there exists h1(z), h2(z) ∈ eP (β) such thath(z) =

µk

4+1

2

¶h1(z)−

µk

4− 12

¶h2(z).

We now introduce a subclass of A as follows.

63

Definition 3.1.2

Let f(z) ∈ A. Then, f(z) ∈ eRk(γ1, γ2, β), k ≥ 2, γ1, γ2 ≥ 0, (γ1 + γ2) > 0, 0 < β ≤ 1, if

and only if ∙γ1

γ1 + γ2f 0(z) +

γ2γ1 + γ2

(zf 0(z))0¸∈ ePk(β).

We note the following.

(i). eR2(γ1, 0, 1) = R, the class of functions with bounded turning.

(ii). eR2(γ1, γ2, 1) = R( γ2γ1+γ2

), the class introduced and studied by Chichra [14].

3.1 Certain properties of the class eRk(γ1, γ2, β)

The main object of this chapter is to investigate some properties of the class eRk(γ1, γ2, β).

Some applications involving integral operators are also considered.

To prove our main results, we need the following.

Lemma 3.1.1[39]

Let h(z) be analytic in E with h(0) = 1 and h(z) 6= 0 in E and suppose that

|arg [h(z) + γzh0(z)]| < π

2

µγ1 +

2

πtan−1 γ1γ2

¶, γ1 > 0, γ2 > 0.

Then

|arg h(z)| < γ1π

2for z ∈ E.

We prove the following results.

Theorem 3.1.1

eRk(γ1, γ2, β) ⊂ eRk(γ1, 0, β1),

where

β = β1 +2

πtan−1

µβ1γ2

γ1 + γ2

¶. (3.1.1)

64

Proof. Set

f 0(z) = h(z) =

µk

4+1

2

¶h1(z)−

µk

4− 12

¶h2(z), z ∈ E,

where h(z) is analytic in E with h(0) = 1. Then

∙h(z) +

γ2γ1 + γ2

zh0(z)

¸∈ ePk(β).

This implies that ∙hi(z) +

γ2γ1 + γ2

zh0i(z)

¸∈ eP (β), i = 1, 2.

Now, with β = β1 +2πtan−1

³β1γ2γ1+γ2

´, we apply Lemma 3.1.1 to have hi(z) ∈ eP (β1),

i = 1, 2. Consequently h(z) ∈ ePk(β1) and f(z) ∈ eRk(γ1, 0, β1) in E. This completes the

proof. ¥

Theorem 3.1.2

Let f(z) ∈ A and let

f 0(z)

µf(z)

z

¶µ−1∈ ePk(γ1 +

2

πtan−1

γ1µ).

Then µf(z)

z

¶µ

∈ ePk(γ1), z ∈ E.

Proof. Let for µ > 0,³f(z)z

´µ= h(z). We note that h(z) is analytic and h(0) = 1. Then

h(z) +1

µzh0(z) = f 0(z)

µf(z)

z

¶µ−1∈ ePk(γ1 +

2

πtan−1

γ1µ).

This implies that for z ∈ E and i = 1, 2

∙hi(z) +

1

µzh0i(z)

¸∈ eP (γ1 + 2π tan−1 γ1µ ),65

and using Lemma 3.1.1, we have hi(z) ∈ eP (γ1). Therefore, by Definition 3.1.1,h(z) ∈ ePk(γ1) and this proves our result. ¥

Theorem 3.1.3

Let

F (z) = Iµ,c(f(z)) =

⎡⎣(µ+ c)

zc

zZ0

tc−1fµ (t) dt

⎤⎦ 1µ

, (3.1.2)

where µ > 0, c+ µ > 0 and Iµ,c(f(z))z

6= 0 in E. Let, for γ1 > 0,

f 0(z)

µf(z)

z

¶µ−1∈ ePk(γ1 +

2

πtan−1

γ1µ+ c

).

Then

F 0(z)

µF (z)

z

¶µ−1∈ ePk(γ1), z ∈ E.

Proof. Let

h(z) = F 0(z)

µF (z)

z

¶µ−1, z ∈ E. (3.1.3)

From (3.1.2), we can write

zcF µ(z) = (µ+ c)

zZ0

tc−1fµ (t) dt. (3.1.4)

Differentiating (3.1.4) and simplifying, we have

cF µ(z) + µzF 0(z)F µ−1(z) = (µ+ c) fµ(z).

This can be written as

c

µF (z)

z

¶µ

+ µzF 0(z)

µF (z)

z

¶µ−1= (µ+ c)

µf(z)

z

¶µ

,

66

that is

c

µF (z)

z

¶µ

+ µh(z) = (µ+ c)

µf(z)

z

¶µ

. (3.1.5)

Differentiating (3.1.5) and using (3.1.3), we have

h(z) +1

µzh0(z) = f 0(z)

µf(z)

z

¶µ−1∈ ePk(γ1 +

2

πtan−1

γ1µ+ c

).

and now proceeding as in Theorem 3.1.2, we obtain the required result. ¥

Theorem 3.1.4

Let f1(z) ∈ eR2(γ1, γ2, β), f2(z) ∈ eRk(γ1, γ2, β) and let, φ(z) = (f1 ∗ f2)(z). Then

(zφ0(z))0

φ0(z)∈ ePk(β3),

where β3 = β1 + β2 and β1, β2 are given by (3.1.1) and (3.1.6). ¤Proof. Since f2(z) ∈ eR2(γ1, γ2, β), it follows from Theorem 3.1.1, f 02(z) ∈ ePk(β1), where

β1 is given by (3.1.1). Similarly f01(z) ∈ eP (β1). Let

f 02(z) = h(z) =

µk

4+1

2

¶h1(z)−

µk

4− 12

¶h2(z),

f 01(z) = h3(z), h1(z), h2(z), h3(z) ∈ eP (β)Now

φ0(z) + zφ00(z) = (f 01 ∗ f 02)(z)

=

µk

4+1

2

¶((h3 ∗ h1)(z))−

µk

4− 12

¶((h3 ∗ h2)(z)).

Applying Lemma 3.1.1, we have φ0(z) ∈ eP (β2), whereβ1 = β2 +

2

πtan−1 β2. (3.1.6)

67

Let H(z) = (zφ0(z))0

φ0(z) . Then

|argH(z)| ≤ |arg(zφ0(z))0|+ |arg φ0(z)|

<k

2

µβ1π

2

¶+

k

2

µβ2π

2

¶=

k

2(β1 + β2)

π

2.

This implies that H(z) ∈ ePk(β3) and the proof is complete. ¥

Theorem 3.1.5

Let f(z) ∈ eRk(γ1, γ2, β) and let ψ(z) be convex univalent function.

Then (ψ ∗ f)(z) ∈ eR2(γ1, γ2, β) in E. ¤Proof. Let

γ1γ1 + γ2

(ψ ∗ f)0(z) + γ2γ1 + γ2

(z(ψ ∗ f)0(z))0

=ψ(z)

z∗∙

γ1γ1 + γ2

f 0(z) +γ2

γ1 + γ2(zf 0(z))0

¸=

ψ(z)

z∗ F (z), F (z) ∈ ePk(β).

We can write

ψ(z)

z∗ F (z) =

µk

4+1

2

¶µψ(z)

z∗ F1(z)

¶−µk

4− 12

¶µψ(z)

z∗ F2(z)

¶.

Since ψ(z) is convex, Reψ(z)z

> 12in E, see [19], and Fi(z) ∈ eP (β), i = 1, 2. Therefore

by Lemma 1.7.2,³ψ(z)z∗ Fi(z)

´lies in the convex hull of Fi(z). Since Fi(z), i = 1, 2, is

analytic in E and Fi(z) ⊂ Ω ≡©wi(z) : |argwi(z)| < βπ

2

ª, it follows that

³ψ(z)z∗ Fi(z)

´lies in Ω. It implies that

³ψ(z)z∗ F (z)

´∈ ePk(β) and consequently (ψ∗f)(z) ∈ eR2(γ1, γ2, β)

in E. ¥The applications of the above theorem are similar as given in Theorem 2.3.1. We now

68

study the converse of Theorem 3.1.1 as follows.

Theorem 3.1.6

Let f(z) ∈ eRk(γ1, 0, β). Then f(z) ∈ eRk(γ1, γ2, β) for |z| < rγ3, where

rγ3 =1

2γ3 +p4γ23 − 2γ3 + 1

, γ3 6=1

2, γ3 =

γ2γ1 + γ2

. (3.1.7)

Proof. Let

φγ3(z) = (1− γ3)f0(z) + γ3(zf

0(z))0

=γ1

γ1 + γ2f 0(z) +

γ2γ1 + γ2

(zf 0(z))0.

Then

φγ3(z) =ψγ3(z)

z∗ f 0(z)

=

µk

4+1

2

¶µψγ3(z)

z∗ h1(z)

¶−µk

4− 12

¶µψγ3(z)

z∗ h2(z)

¶, (3.1.8)

where

ψγ3(z) = (1− γ3)

z

1− z+ γ3

z

(1− z)2.

Now ψγ3(z) is convex in |z| < rγ3 , which implies that Re

ψγ3 (z)

z> 1

2for |z| < rγ3 and rγ3

is given by (3.1.7). Therefore, from Lemma 1.7.2,³ψγ3(z)

z∗ hi(z)

´, i = 1, 2 lies in the

convex hull hi(E) in |z| < rγ3 . Since hi(z) is analytic in E and

Ω ≡½wi(z) : |argwi(z)| <

βπ

2

¾,

³ψγ3 (z)

z∗ hi(z)

´lies in Ω for |z| < rγ3 . It implies that

³ψγ3 (z)

z∗ h(z)

´belongs to ePk(β)

for |z| < rγ3 and consequently f(z) ∈ eRk(γ1, γ2, β) for |z| < rγ3. ¥

69

Chapter 4

On a certain class of analytic

functions and Hankel determinant

problem

The class Nk discussed before in chapter one was first introduced by Noor [64] as a gen-

eralization of close-to-covexity . She studied its geometrical interpretation and various

other interesting properties including growth rate of coefficient differences and radius

of convexity problem. Recently, Noor [67], studied the class of analytic functions corre-

sponding to strongly close-to-convex functions. She employed a modification to a method

of Pommerenke [96] to investigate the growth rate of Hankel determinant problems re-

garding this class.

In this chapter, we define a class of analytic function related with strongly close-to-

convex functions. We shall investigate different interesting properties including inclusion

relations, arc length, growth rate of coefficients and growth rate of Hankel determinant

by using a different method from that of Noor [67].We observed that certain well-known

results come out as a special case from our results signifying the work presented here in

this chapter.

70

We now define the following classes of analytic functions.

Definition 4.1.1

Let f(z) ∈ A be locally univalent in E. Then, for η 6= 0 (complex), 0 ≤ ρ < 1,

f(z) ∈ Vk(η, ρ), if and only ifµ1 +

1

η

zf 00(z)

f 0(z)

¶∈ Pk(ρ), z ∈ E.

We note that for η = 1, we have the class Vk(ρ) introduced by Padmanabhan and

Parvatham [91].

Definition 4.1.2

Let f(z) ∈ A. Then f(z) ∈ eNk(η, ρ, β), if and only if, for k ≥ 2, 0 < β ≤ 1, there exists

a function g(z) ∈ Vk(η, ρ) such that¯arg

f 0(z)

g0(z)

¯≤ βπ

2, z ∈ E.

For η = 1, this class was recently introduced and studied by Noor [67]. For k = 2, η = 1,

ρ = 0, eN2(1, 0, β) is the class of strongly close-to-convex functions. Also eN2(1, ρ, 0) =

C(ρ) is class of convex functions of order ρ. For η = 1, ρ = 0, β = 1, the class ofeNk(η, ρ, β) reduces to the class Nk introduced by Noor [64].

We need the following results in our investigations.

Lemma 4.1.1

A function f(z) ∈ Vk(η, ρ), if and only if

(i). f 0(z) = [f 01(z)](1−ρ)η , f1(z) ∈ Vk,

(ii). f 0(z) = [f 02(z)]η , f2(z) ∈ Vk(ρ),

71

(iii). there exists two normalized starlike functions s1(z) and s2(z) such that

f 0(z) =

"(s1(z)/z)

(k4+12)

(s2(z)/z)(k4−12)

#(1−ρ)η. (4.1.1)

The above lemma is special case of a result discussed in [5]. ¤

4.1 Some properties of the class eNk(η, ρ, β)

Following essentially the same method due to Noor, we can easily derive the following

result. We include the details for the sake of completeness.

Theorem 4.1.1

The function f(z) ∈ eNk(η, ρ, β), if and only if

f 0(z) =(f1(z))

(k4+ 12)(1−ρ)η

(f2(z))(k4− 12)(1−ρ)η

,

where f1(z) and f2(z) are strongly close-to-convex functions of order β. ¤Proof. From Definition 4.1.2, we have

f 0(z) = g0(z)hβ(z), g(z) ∈ Vk(η, ρ), h(z) ∈ P.

Now from Lemma 4.1.1, we have

g0(z) =(g01(z))

(k4+12)(1−ρ)η

(g02(z))(k4−12)(1−ρ)η

, g1(z), g2(z) ∈ C.

Thus

f 0(z) =(g01(z))

(k4+12)(1−ρ)η

(g02(z))(k4−12)(1−ρ)η

hβ(z) =(f1(z))

(k4+ 12)(1−ρ)η

(f2(z))(k4− 12)(1−ρ)η

.

This completes the proof. ¥

72

Theorem 4.1.2

Let f(z) ∈ eNk(η, ρ, β) in E. Then f(z) ∈ Cη for |z| < r0, where

r0 =[(1− ρ) |η| k + 2β]−

q[(1− ρ) |η| k + 2β]2 − 4(1− 2ρ) |η|2

2(1− 2ρ) |η| . (4.1.2)

This result is sharp. ¤Proof. We can write

f 0(z) = g0(z)hβ(z), g(z) ∈ Vk(η, ρ), h(z) ∈ P.

Now using Lemma 4.1.1, we have

f 0(z) =

"(s1(z)/z)

(k4+12)

(s2(z)/z)(k4−12)

#(1−ρ)ηhβ(z), (4.1.3)

where s1(z) and s2(z) are starlike functions. Logarithmic differentiation of (4.1.3) gives

uszf 00(z)

f 0(z)= (1− ρ)η

∙−1 +

µk

4+1

2

¶zs01(z)

s1(z)−µk

4− 12

¶zs02(z)

s2(z)

¸+ β

zh0(z)

h(z),

implies that

1 +1

η

zf 00(z)

f 0(z)= ρ+ (1− ρ)

∙µk

4+1

2

¶zs01(z)

s1(z)−µk

4− 12

¶zs02(z)

s2(z)

¸+

β

η

zh0(z)

h(z).

Now using Theorem 1.3.2, we have

Re

µ1 +

1

η

zf 00(z)

f 0(z)

¶≥ ρ+ (1− ρ)

∙µk

4+1

2

¶1− r

1 + r−µk

4− 12

¶1 + r

1− r

¸− β

|η|2r

1− r2

=ρ |η| (1− r2) + (1− ρ) |η| [1− kr + r2]− 2βr

|η| (1− r2). (4.1.4)

73

The right hand side of (4.1.4) is positive for |z| < r0, where r0 is given by (4.1.2). The

sharpness can be viewed from the function f0(z) ∈ eNk(η, ρ, β), given by

f 00(z) =(1 + z)(

k2−1)(1−ρ)η+β

(1− z)(k2+1)(1−ρ)η+β

, z ∈ E. (4.1.5)

We note that:

(i). For η = 1, we have the radius of convexity for the class fNk(ρ, β) studied by Noor

[67].

(ii). For η = 1, ρ = 0, β = 1, we have the radius of convexity for the class Nk, proved

by Noor [64].

(iii). For η = 1, ρ = 0, β = 1, k = 2, we have radius of convexity for close-to-convex

functions which is well-known.

We now discuss the arc length problem and growth of coefficients for the classfNk(η, ρ, β).

Theorem 4.1.3

Let f(z) ∈ fNk(η, ρ, β), for Re η > 0, 0 < β ≤ 1, 0 ≤ ρ < 1 and (k+2)(1−ρ)Re η2−β > 1. Then

Lr(f) ≤ c(k, η, ρ, β)

µ1

1− r

¶(k2+1)(1−ρ)Re η+β−1

,

where c(k, η, ρ, β) is a constant depending only on k, η, ρ, β.The exponent£(k2+ 1)(1− ρ)Re η + β − 1

¤is sharp. ¤Proof. We have

Lr(f) =

2πZ0

|zf 0(z)| dθ, z = reiθ.

74

Using the Definition 4.1.2, Lemma 4.1.1 (iii) and Theorem 1.4.3, we have

Lr(f) =

2πZ0

¯zg0(z)hβ(z)

¯dθ, g(z) ∈ Vk(η, ρ), h(z) ∈ P

=

2πZ0

¯¯z (s1(z)/z)(

k4+ 12)(1−ρ)η

(s2(z)/z)(k4− 12)(1−ρ)η

¯¯ ¯hβ(z)¯ dθ

=

2πZ0

¯¯z1−η(1−ρ) (s1(z))(

k4+ 12)(1−ρ)η

(s2(z))(k4− 12)(1−ρ)η

¯¯ ¯hβ(z)¯ dθ

≤ 2(k2−1)(1−ρ)Re η

r(k4+ 12)(1−ρ)Re η−1

2πZ0

|s1(z)|(k4+12)(1−ρ)Re η |h(z)|β dθ.

Using Holder’s inequality with p = 22−β , q =

2βsuch that 1

p+ 1

q= 1, we obtain

Lr(f) ≤2(

k2−1)(1−ρ)Re η

r(k4+ 12)(1−ρ)Re η−1

⎛⎝ 1

2π

2πZ0

|s1(z)|(k2+1)(1−ρ) Re η

2−β dθ

⎞⎠2−β2⎛⎝ 1

2π

2πZ0

|h(z)|2 dθ

⎞⎠β2

.

Since (k+2)(1−ρ)Re η2−β > 1, therefore we use subordination for starlike functions and Lemma

1.3.2, to have

Lr(f) ≤ c(k, η, ρ, β)

µ1

1− r

¶(k2+1)(1−ρ)Re η+β−1

.

The function F0(z) ∈ fNk(η, ρ, β) defined by

F 00(z) = G0

0(z)hβ0(z), (4.1.6)

where

G00(z) =

(1 + z)(k2−1)(1−ρ)η

(1− z)(k2+1)(1−ρ)η

and h0(z) =1 + z

1− z,

shows that the exponent is sharp. ¥By assigning different values to the parameters involved in the above theorem, we have

75

the following interesting results.

Corollary 4.1.1

Let f(z) ∈ fNk(ρ, β). Then

Lr(f) ≤ c(k, ρ, β)

µ1

1− r

¶(k2+1)(1−ρ)+β−1

.

Corollary 4.1.2

Let f(z) ∈ Nk. Then

Lr(f) ≤ c(k, ρ, β)

µ1

1− r

¶k2+1

.

Coefficient growth problems

The problem of growth and asymptotic behavior of coefficients is well-known. In the

upcoming results, we investigate these problems for different set of classes by varying

different parameters.

Theorem 4.1.4

Let f(z) ∈ fNk(η, ρ, β) and be of the form (1.2.1). Then, for n > 3, k ≥ 2, Re η > 0,

0 ≤ ρ ≤ 1, 0 < β ≤ 1, we have

|an| ≤ c(k, η, ρ, β)n(k2+1)(1−ρ)Re η+β−2.

where c(k, η, ρ, β) is a constant depending only on k, η, ρ, β.The exponent£(k2+ 1)(1− ρ)Re η + β − 2

¤is sharp. ¤Proof. With z = reiθ, Cauchy’s theorem gives us

nan =1

2πrn

2πZ0

|zf 0(z)| dθ = 1

2πrnLr(f), z = reiθ.

76

Using Theorem 4.1.3 and putting r = 1− 1n, we obtain the required result. The sharpness

follows from the function F0(z) defined by the relation (4.1.6).

Corollary 4.1.3

Let f(z) ∈ eNk(ρ, β) and be of the form (1.2.1). Then, for n > 3, k ≥ 2, we have

|an| = O(1)n(k2+1)(1−ρ)+β−2.

For ρ = 0, β = 1 in the above corollary, we have growth rate of coefficients problem

for functions in this class Nk and for k = 2, ρ = 0, β = 1 gives us the growth rate of

coefficient estimates for close-to-convex functions, which is well-known.

4.2 Hankel determinant problem

The Hankel determinant of a function f(z) of the form (1.2.1) is defined by

Hq(n) =

¯¯¯an an+1 . . . an+q−1

an+1 an+2 . . . an+q...

......

...

an+q−1 an+q . . . an+2q−2

¯¯¯ . (4.2.1)

The Hankel determinant occupies a central place, such as, in the discussion of singularities

by Hadamarad [18, p 329], Polya, Edrei [20] and in the investigation of power series with

integral coefficients by Polya [18, p 329], Cantor [12] and many others. The rate of growth

of Hankel determinant Hq(n) as n→∞, when f(z) is a member of any class of analytic

function is well-known. Pommerenke [99], proved that when f is areally-mean p-valent

function, then for p ≥ 1

Hq(n) = O(1)ns√q− q

2 , as n→∞,

77

and s = 16p3/2 and where O(1) depends on p, q, and the function f(z). In particular,

this shows that Hq(n)→ 0 as n→∞ for large q relative to p. In fact for p = 1, q = 2

H2(n) = O(1)n12 , as n→∞.

The exponent 12is exact.

Noonan and Thomas [58] gave the exact rate of growth of Hq(n) for large p relative

to q, and they proved that

Hq(n) = O(1)

⎧⎨⎩ n2p−1, q = 1, p > 14,

n2pq−q2, q ≥ 2, p ≥ 2(q − 1),

where O(1) depends upon p, q only and the exponent 2pq− q2 is best possible. Also, for

univalent functions, Pommerenke [96] has proved that for q ≥ 2,

Hq(n) < c(q) n−(12+β)q+3

2 , (n→∞) ,

where β > 14000

, which in particular shows that

H2(n) < c n12−2β.

Pommerenke [99] has shown that, if f(z) is starlike, then for q ≥ 1,

Hq(n) = O(1) n2−q, (n→∞),

where O(1) depends upon q only and the exponent 2 − q is best possible. Noor [69]

generalized this result for close-to-convex functions. We also refer to [67] and [70, 78].

Also, for f(z) ∈ Vk, it is shown [61] that for q ≥ 1, n→∞,

Hq(n) = O(1)

⎧⎨⎩ nk2−1, q = 1,

nkq2−q2, q ≥ 2, k ≥ 8q − 10,

78

where O(1) depends upon p, q and f(z) only. The exponent kq2− q2 is best possible.

In [66], it is proved that for f(z) ∈ Nkk,

Hq(n) = O(1)

⎧⎨⎩ nk2 , q = 1

nkq2−q2+q, q ≥ 2, k ≥ 8q − 10.

Definition 4.2.1

Let z1 be a non-zero complex number. Then for f(z), given by (1.2.1), we define

∆j (n, z1, f(z)) = ∆j−1 (n, z1, f(z))− z1∆j−1 (n+ 1, z1, f(z)) , j ≥ 1,

with ∆1 (n, z1, f(z)) = an.

The following two lemmas are due to Noonan and Thomas [58] and we include their

proofs for the sake of completeness.

Lemma 4.2.1

Let f(z) ∈ A and let the Hankel determinant of f(z) be defined by (4.2.1). Then, writing

∆j = ∆j (n, z1, f(z)), we have

Hq(n) =

¯¯¯∆2q−2(n) ∆2q−3(n+ 1) . . . ∆q−1(n+ q − 1)

∆2q−3(n+ 1) ∆2q−4(n+ 2) . . . ∆q−2(n+ q)...

......

...

∆q−1(n+ q − 1) ∆q−2(n+ q) . . . ∆0(n+ 2q − 2)

¯¯¯ . (4.2.2)

Proof. For j = 2, . . . , q, we multiply jth row in (4.2.1) by z1 and subtract from (j − 1) th

row. In the resulting determinant, for j = 2, . . . , q − 1, we again multiply jth row by z1and subtract from (j − 1) th row. Using Definition 4.2.1, we repeat this process until we

79

have

Hq(n) =

¯¯¯∆q−1(n) ∆q−1(n+ 1) . . . ∆q−1(n+ q − 1)

∆q−2(n+ 1) ∆q−2(n+ 2) . . . ∆q−2(n+ q)...

......

...

∆0(n+ q − 1) ∆0(n+ q) . . . ∆0(n+ 2q − 2)

¯¯¯ . (4.2.3)

Next we operate on the columns of (4.2.3) in the following way. Let Ci, i = 1, 2, . . . , q

be the ith column in (4.2.3). Define, for j = 1, . . . , q − 1, the vector

Dq−j =

jXl=1

(−1)l−1µj

l

¶zl1 Cq−j+l,

and replace the Cq−jth column in (4.2.3) by Cq−j − Dq−j. Continuing this process for

j = 1, . . . , q − 1, we obtain (4.2.2). ¥

Lemma 4.2.2

With z1 =n

n+1y, and v ≥ 0 any integer

∆j (n+ v, z1, zf0(z)) =

jXl=0

µj

l

¶yl (v − (l − 1)n)

(n+ 1)l∆j−l (n+ v + l, y, f(z)) . ¤

Proof. It follows easily by induction that

∆j (n+ v, z1, zf0(z)) =

jXl=0

(−1)lµj

l

¶zl1 (n+ v + l) an+v+l, (4.2.4)

80

and using z1 = nn+1

y, we have

jXl=0

µj

l

¶yl (v − (l − 1)n)

(n+ 1)l∆j−l (n+ v + l, y, f(z))

=

jXl=0

j−lXi=0

µl

j

¶µj − l

i

¶(n+ 1)i

nl+izl+i1 (−1)i (v − (l − 1)n) an+v+l+i.

(4.2.5)

The proof is complete if we show that, for 0 ≤ m ≤ j, the coefficient of an+v+m in (4.2.5)

is equal to the corresponding coefficient in (4.2.4). The coefficient of an+v+m in (4.2.4) is

(−1)mµl

m

¶zm1 (n+ v +m),

and the coefficient in (4.2.5) is, since l + i = m,

mXl=0

¡jl

¢¡j−mm−l¢(n+1)m−l

nmzm1 (−1)m−l (v − (l − 1)n)

=¡n+1n

¢m(−1)m zm1

mXl=0

¡jl

¢¡j−lm−l¢ ¡ −1

n+1

¢l(v − (l − 1)n).

Since µj

m

¶µj − l

m− j

¶=

µj

m

¶µm

l

¶,

it suffices to show that

(n+ v +m) =

µn+ 1

n

¶m mXl=0

µm

l

¶µ−1n+ 1

¶l

(v − (l − 1)n) .

However, this follows relatively easily by induction and the proof is complete. ¥We shall also need the following remark given in [58].

81

Remark 4.2.1

Consider any determinant of the form

D =

¯¯¯y2q−2 y2q−3 . . . yq−1

y2q−3 y2q−4 . . . yq−2...

......

...

yq−1 yq−2 . . . y0

¯¯¯ ,

with 1 ≤ i, j ≤ q and αi,j = y2q−(i+j) , D = det (αi,j). Thus

D =X

v1∈ Sq

(sgn v1)

qYj=1

y2q−(v1(j)+j),

where Sq is the symmetric group on q elements, and sgn v1 is either +1 or −1. Thus, in

the expansion of D, each summand has q factors, and the sum of the subscripts of the

factors of each summand is q2 − q.

Now let n be given and Hq(n) is as in Lemma 4.2.1, then each summand in the expansion

of Hq(n) is of the formqY

i=1

∆v1(i) (n+ 2q − 2− v1(i)) ,

where v1 ∈ Sq andqX

i=1

v1 (i) = q2 − q ; 0 ≤ v1(i) ≤ 2q − 2.

We now prove the following.

Theorem 4.2.1

Let f(z) ∈ eNk(η, ρ, β) and let the Hankel determinant of f(z), for q ≥ 2, n ≥ 1, be

defined by (4.2.1). Then, for q ≥ 2 and k > 4 (q−1)(1−ρ)Re η − 2, we have

Hq(n) = O(1) n[(k2+1)(1−ρ)Re η+β]q−q2 ,

82

where O(1) depends only on k, η, ρ, β and q.

Proof. Since f(z) ∈ eNk(η, ρ, β), there exists g(z) ∈ Vk(η, ρ) such that

f 0(z) = g(z) hβ(z) ∈ P , z ∈ E.

Now, for j ≥ 1, z1 any non-zero complex number and z = reiθ, we consider for F (z) =

zf 0(z),

|∆j (n, z1, F (z))| =

¯¯ 1

2πrn+j

2πZ0

(z − z1)j F (z)e−i(n+j)θdθ

¯¯

≤ 1

2πrn+j

2πZ0

|z − z1|j¯¯z1−(1−ρ)η (s1(z)/z)(

k4+ 12)(1−ρ)η

(s2(z)/z)(k4− 12)(1−ρ)η

¯¯ ¯hβ(z)¯ dθ

≤ 1

2πrn+j

2πZ0

|z − z1|j |s1(z)|j|s1(z)|(

k4+ 12)(1−ρ)Re η−j

|s2(z)|(k4− 12)(1−ρ)Re η

|h(z)|β dθ,

where we have used Theorem 4.1.1.Using Lemma 1.4.1, we have

|∆j (n, z1, F (z))| ≤1

2πrn+j

µ2r2

1− r2

¶j2πZ0

|s1(z)|(k4+ 12)(1−ρ)Re η−j

|s2(z)|(k4−12)(1−ρ)Re η

|h(z)|β dθ. (4.2.6)

Now using Theorem 1.4.3 and simplifying, we obtain from (4.2.6),

|∆j (n, z1, F (z))| ≤1

2π

(2)(k2−1)(1−ρ)Re η

r(k4− 12)(1−ρ)Re η+n−j−1

µ1

1− r

¶j2πZ0

|s1(z)|(k4+ 12)(1−ρ)Re η−j |h(z)|β dθ.

83

Using Holder’s inequality, with p = 22−β , q =

2β, such that 1

p+ 1

q= 1, we can write

|∆j (n, z1, F (z))| ≤(2)(

k2−1)(1−ρ)Re η

r(k4− 12)(1−ρ)Re η+n−j−1

µ1

1− r

¶j⎛⎝ 1

2π

2πZ0

|s1(z)|(k2+1)(1−ρ) Re η−2j

2−β

⎞⎠2−β2

⎛⎝ 1

2π

2πZ0

|h(z)|2 dθ

⎞⎠β2

.

Now proceeding in a similar way as in Theorem 4.1.3, we have

|∆j (n, z1, F (z))| ≤(2)(

k2−1)(1−ρ)Re η+β

2

rn−1

µ1

1− r

¶β2+j⎛⎝ 1

2π

2πZ0

1

|1− reiθ|(k+2)(1−ρ) Re η−4j

2−βdθ

⎞⎠2−β2

.

Now subordination for starlike functions yields us

|∆j (n, z1, F (z))| = O(1)µ

1

1− r

¶(k2+1)(1−ρ)Re η−j+β−1

,

where O(1) depends only on k, η, β and j.

Now applying Lemma 4.2.2 and putting z1 =n

n+1eiθn , (n→∞), we have for k ≥³

4j(1−ρ)Re η − 2

´, j ≥ 1

∆j

¡n, eiθn, f(z)

¢= O(1) n(

k2+1)(1−ρ)Re η−j+β−1.

We now estimate the rate of growth of Hq(n). For q = 1, Hq(n) = an = ∆0(n) and from

Theorem 4.1.4, it follows that

H1(n) = O(1) n(k2+1)(1−ρ)Re η+β−2.

84

For q ≥ 2, we use Remark 4.2.1 together with Lemma 4.2.1, to have

Hq(n) = O(1) nq[(k2+1)(1−ρ)Re η+β]−q2, k >

µ4(q − 1)

(1− ρ)Re η− 2¶,

where O(1) depends only on k, η, ρ, β and q. ¥By giving special values to the parameters involved in the above theorem, we obtain the

following interesting results.

Corollary 4.2.1

Let f(z) ∈ eNk(ρ, β) and be defined as in (1.2.1). Then, for q ≥ 2, k >³4(q−1)1−ρ − 2

´,

Hq(n) = O(1) nq[(k2+1)(1−ρ)+β]−q2 , (n→∞),

where O(1) depends only on k, ρ, β and q. ¤Noor [67] studied the above corollary with a different method.

Corollary 4.2.2

Let f(z) ∈ Nk and be defined as in (1.2.1). Then, for q ≥ 2, k > (4q − 6) ,

Hq(n) = O(1) nq[(k2+2]−q2, (n→∞),

where O(1) depends only on k and q. ¤

85

Chapter 5

Some applications of Ruscheweyh

derivatrives

This chapter is mainly concerned with the analytic functions which are defined by means

of Ruschweyh derivatives. The classes of functions with bounded boundary and bounded

radius rotations along with functions with bounded Mocanu variations are studied under

the Ruschweyh derivatives. These classes are special generalizations of convex and related

functions. The class of functions with bounded boundary was first taken into account

by Paatero [89] though it was originally introduced by Lowner [45]. Paatero [89] widely

studied its basic properties and developed the theory of this class. In 1952, Tammi [121]

introduced the class of functions with bounded radius rotation. Both of these classes were

considered by Padmanabhan and Parvatham [91] by using the order terminology. The

concept of bounded Mocanu variations was first introduced and investigated by Coonce

and Ziegler [17]. The class of such functions actually provides a connection between the

functions with bounded boundary and bounded radius rotations, see for recent work [79,

80]. These concepts shall be investigated along with some of their interesting properties.

The results such as inclusion relations, some basic properties involving generalized

Bernardi integral operator and radius problems are studied by using the convolution

technique [68, 79] and the properties of the class P. Many insightful interesting results

86

are observed as special cases signifying the work presented here.

5.1 Certain analytic functions defined by Rusheweyh

derivatives

In this section, we define certain classes of analytic functions defined by Ruscheweyh

derivatives and study some of their properties. We have the following.

Definition 5.1.1

Let f(z) ∈ A. Then f(z) ∈ Rk(m, ρ), k ≥ 2, m ∈ N = 1, 2, 3, . . ., 0 ≤ ρ < 1, if and

only if, Dm+1f(z)Dmf(z)

∈ Pk(ρ) for z ∈ E.

Definition 5.1.2

Let f(z) ∈ A. Then f is said to belong to the classMk,γ(m, ρ) if and only if for f(z)f 0(z) 6=

0 in 0 < |z| < 1 and for γ ≥ 0, k ≥ 2, m ∈ N = 1, 2, 3...., 0 ≤ ρ < 1,

Jm(f(z), γ) ∈ Pk(ρ), where

Jm(f(z), γ) = (1− γ)Dm+1f(z)

Dmf(z)+ γ

Dm+2f(z)

Dm+1f(z). (5.1.1)

We note that, for k = 2, β = 12, R2(m, 1

2) was defined by Ruscheweyh [112], and

M2,γ(m, 12) was introduced and discussed in [2].

5.1.1 Some inclusion problems and integral preserving property

The inclusion problems between different new as well as the classical classes of analytic

functions is well-known. The inclusion properties between the above mentioned classes

along with the preservance under Bernardi integral transform will be investigated here.

87

Theorem 5.1.1

For 1m+2≤ ρ0 ≤ 1, ρ(m+ 1) = ρ0(m+ 2)− 1

Rk(m+ 1, ρ0) ⊂ Rk(m, ρ).

Proof. The following identity can easily be verified

z (Dmf (z))0 = (m+ 1)Dm+1f (z)−mDmf (z) . (5.1.2)

Let f(z) ∈ Rk(m+ 1, ρ0). SetDm+1f(z)

Dmf(z)= h0(z), (5.1.3)

h0(z) is analytic in E and h0(0) = 1. Differentiating (5.1.3) and using (5.1.2), we have

(m+ 2)Dm+2f(z)

Dm+1f(z)= (m+ 1)h0(z) + 1 +

zh00(z)

h0(z).

Let h0(z) = (1− ρ)h(z) + ρ and

h (z) =

µk

4+1

2

¶h1 (z)−

µk

4− 12

¶h2 (z) . (5.1.4)

Then

Dm+2f(z)

Dm+1f(z)=(m+ 1)(1− ρ)

(m+ 2)h(z) +

(m+ 1)ρ+ 1

(m+ 2)+

1

m+ 2

(1− ρ)zh0(z)

(1− ρ)h(z) + ρ.

That is

Dm+2f(z)

Dm+1f(z)− (m+ 1)ρ+ 1

(m+ 2)=(m+ 1)(1− ρ)

(m+ 2)h(z) +

1

m+ 2

(1− ρ)zh0(z)

(1− ρ)h(z) + ρ.

88

With ρ0 =(m+1)ρ+1(m+2)

, we have

1

1− ρ0

∙Dm+2f(z)

Dm+1f(z)− ρ0

¸= h(z) +

1

m+ 1

zh0(z)

(1− ρ)h(z) + ρ

= h(z) +

1(m+1)(1−ρ)zh

0(z)

h(z) + ρ1−ρ

.

Let a = 1(m+1)(1−ρ) , b =

ρ1−ρ and define

ϕa,b(z) =1

1 + b

z

(1− z)a+1+

b

b+ 1

z

(1− z)a+2. (5.1.5)

Using the convolution technique [68, 79] together with (5.1.4), we have

h(z) ∗ϕa,b(z)

z= h(z) +

azh0(z)

h(z) + b

=

µk

4+1

2

¶µh1(z) ∗

ϕa,b(z)

z

¶−µk

4− 12

¶µh2(z) ∗

ϕa,b(z)

z

¶=

µk

4+1

2

¶µh1(z) +

azh01(z)

h1(z) + b

¶−µk

4− 12

¶µh2(z) +

azh02(z)

h2(z) + b

¶.

Since f(z) ∈ Rk(m+ 1, ρ0), therefore

1

1− ρ0

∙Dm+2f(z)

Dm+1f(z)− ρ0

¸∈ Pk, (z ∈ E)

and so³h(z) + azh0(z)

h(z)+b

´∈ Pk and this implies that

µhi(z) +

azh0i(z)

hi(z) + b

¶∈ P, z ∈ E, i = 1, 2.

We now form functional Ψ(u, v) by taking u = hi(z), v = zh0i(z). It can be easily seen

that

(i) Ψ(u, v) is continuous in a domain D = C− −b ×C.

(ii) (1, 0) ∈ D and Ψ(1, 0) = 1 > 0.

89

We now check the condition (iii) of Lemma 1.6.1 as follows

Reψ(iu2, v1) = Reav1

b+ iu2=

abv1b2 + u22

≤ −12

ab(1 + u22)

b2 + u22≤ 0,

since a = 1(m+1)(1−ρ) > 0, b = ρ

1−ρ ≥ 0. Hence, applying Lemma 1.6.1, it follows

that hi(z) ∈ P, i = 1, 2; z ∈ E and consequently h(z) ∈ Pk(ρ). This shows f(z) ∈

Rk(m, ρ), z ∈ E and the proof is complete. ¥

Theorem 5.1.2

For (m− γ + 2) > 0,

Mk,γ(m, ρ) ⊂ Rk(m,1

2),

where ρ = 12(1 + γ

m+2), γ ≥ 0 and m ∈ N. ¤

Proof. Let Dm+1f(z)Dmf(z)

= 12(h(z) + 1).

Then h(z) is analytic and h(0) = 1, for z ∈ E. Proceeding as in Theorem 5.1.1, we obtain

2(m+ 2)

m− γ + 2

∙Jm(f(z), γ)−

1

2

µ1 +

γ

m+ 2

¶¸= h(z) +

( 2γm−γ+2)zh

0(z)

h(z) + 1.

That gives us, with ρ = m+γ+22(m+2)

,

1

1− ρ[Jm(f(z), γ)− ρ] =

µk

4+1

2

¶"h1(z) +

2γm−γ+2zh

01(z)

h1(z) + 1

#

−µk

4− 12

¶"h2(z) +

2γm−γ+2zh

02(z)

h2(z) + 1

#.

90

Since f(z) ∈Mk,γ(m, ρ),

"hi(z) +

2γm−γ+2zh

0i(z)

hi(z) + 1

#∈ P, i = 1, 2

and using Lemma 1.6.1 as before in Theorem 5.1.1, we have hi(z) ∈ P and this proves

the result. ¥The upcoming result is about the perservance property of the Bernardi integral op-

erator defined by (1.7.1). We prove the following.

Theorem 5.1.3

Let F (z) be defined by (1.7.1) with f(z) ∈ Rk(m, ρ), c > −1 and

max

½m− c

m+ 1,2m− c

2(m+ 1)

¾≤ ρ < 1.

Then F (z) ∈ Rk(m,bρ), where bρ is given as in Lemma 1.6.7. This result is sharp. ¤Proof. From (1.7.1) , we can easily derive the formula

z (DmF (z))0 = (1 + c)Dmf(z)− cDmF (z). (5.1.6)

LetDm+1F (z)

DmF (z)= h(z),

where h(z) is analytic in E with h(0) = 1.From (1.7.3) and (5.1.6) , we have

(1 + c)Dm+1f(z) = cDm+1F (z) + z¡Dm+1F (z)

¢0= c [h(z)DmF (z)] + z [h(z)DmF (z)]0

= c [h(z)DmF (z)] + zh0(z)DmF (z)

+ h(z)£(m+ 1)Dm+1F (z)−mDmF (z)

¤=£(c−m)h(z) + zh0(z) + (m+ 1)h2(z)

¤DmF (z). (5.1.7)

91

Similarly, we have

(1 + c)Dmf(z) = z(DmF (z))0 + cDmF (z)

= [(c−m) + (m+ 1)h(z)]DmF (z). (5.1.8)

Therefore from (5.1.7) and (5.1.8), we obtain

Dm+1f(z)

Dmf(z)= h(z) +

zh0(z)

(m+ 1)h(z) + (c−m),

and since f(z) ∈ Rk(m, ρ), therefore

∙h(z) +

zh0(z)

(m+ 1)h(z) + (c−m)

¸∈ Pk(ρ).

Using again the convolution technique [68, 79] and writing

h(z) =

µk

4+1

2

¶h1 (z)−

µk

4− 12

¶h2 (z) ,

we have,

h(z) +zh0(z)

(m+ 1)h(z) + (c−m)=

ϕa,b(z)

z∗ h(z)

=

µk

4+1

2

¶hh1(z) +

zh01(z)(m+1)h1(z)+(c−m)

i−µk

4− 12

¶hh2(z) +

zh02(z)(m+1)h2(z)+(c−m)

i,

where ϕa,b is defined by (5.1.5) with a = 1m+1

, b = c−mm+1

. This implies that

∙hi(z) +

zh0i(z)

(m+ 1)hi(z) + (c−m)

¸∈ P (ρ), i = 1, 2.

We use Lemma 1.6.7 and it follows that hi(z) ∈ P (bρ), i = 1, 2. Consequently

h(z) ∈ P (bρ) and this completes the proof. ¥92

5.1.2 Various interesting implications

Using the properties of the hypergeometric functions and assigning different values to

m ∈ N, c > −1, we obtain the following interesting special cases of Theorem 5.1.3.

1. Let −1 ≤ c ≤ 0, and ρ = m−cm+1

, then f(z) ∈ Rk(m, ρ) implies F (z) ∈ Rk(m,bρ), wherebρ = 1

(m+ 1)√π

Γ(c+ 32)

Γ(c+ 1)+

m− c

m+ 1,

where Γ denotes Gamma function, and this result is sharp.

2. With c = 0, we have mm+1≤ ρ < 1 and f(z) = zF 0(z) with f(z) ∈ Rk(m, ρ) implies

F (z) ∈ Rk(m,bρ), where

bρ =⎧⎪⎪⎪⎨⎪⎪⎪⎩

1m+1

h1−2(m+1)(1−ρ)2−22(m+1)(1−ρ) +m

i, for ρ 6= 2m+1

2(m+1),

1m+1

£1

2 ln 2+m

¤, for ρ = 2m+1

2(m+1).

This result is sharp. For k = 2, m = 0 it yields a well-known result that a convex

function of order ρ is starlike of order bρ.3. Taking m = 0 in (2), we have 0 ≤ ρ < 1. Then f(z) ∈ Rk(m, ρ) implies

F (z) ∈ Rk(m,bρ), z ∈ E, where

bρ =⎧⎪⎪⎪⎨⎪⎪⎪⎩

2ρ−12−22(1−ρ) , for ρ 6=

12,

12 ln 2

, for ρ = 12.

This result is sharp.

4. Let c > 0, and ρ0 = maxm−cm+1

, 2m−c2(m+1)

= 2m−c2(m+1)

. Then f(z) ∈ Rk(m, ρ) implies

F (z) ∈ Rk(m,bρ) with bρ = 2m−c2(m+1)

. This result is sharp.

5. With c = 1, We have Libra’s integral operator and Theorem 5.1.3 yields several

interesting special cases.

93

6. When c = 0, we obtain the Alexander integral operator defined by (1.4.5), and we

obtain various new and known results for different values of k,m and ρ.

5.1.3 Radius problem

The radius problem or the inverse inclusion of the above result is given here. This means

that when F (z) is taken from the class Rk(m, ρ), then f(z) maps the disk of certain

radius onto the image domain of F (z).

Theorem 5.3.1

Let F (z) be defined by (1.7.1) and for c > m(1 − ρ) + ρ, belong to Rk(m, ρ). Then

f(z) ∈ Rk(m, ρ) in |z| < rm,c is given as

rm,c =|µ+ 1|q

t+ (t2 − |µ2 − 1|2) 12, (5.3.1)

where t = 2(s + 1)2 + |µ|2 − 1, s = 1(m+1)(1−ρ) and µ = c+ρ−m+mρ

(m+1)(1−ρ) 6= −1. This result is

sharp. ¤Proof. From (1.7.1), we can write

f(z) =z1−c

1 + c(zcF (z))0.

With Dm+1F (z)DmF (z)

= h(z) ∈ Pk(ρ), we proceed as in Theorem 5.1.2 and have

Dm+1f(z)

Dmf(z)= h(z) +

zh0(z)

(m+ 1)h(z) + (c−m).

Let h(z) ∈ Pk and

h0(z) = (1− ρ)h(z) + ρ

= (1− ρ)

½µk

4+1

2

¶h1(z)−

µk

4− 12

¶h2(z)

¾+ ρ,

94

where h1(z), h2(z) ∈ P.

Thus using the convolution technique [68, 79] as in Theorem 5.1.3, we have

1

1− ρ

∙Dm+1f(z)

Dmf(z)− ρ

¸= h(z) +

szh0(z)

h(z) + µ

=

µk

4+1

2

¶ ∙h1(z) +

szh01(z)

h1(z) + µ

¸−µk

4− 12

¶ ∙h2(z) +

szh02(z)

h2(z) + µ

¸, (5.3.2)

with s = 1(m+1)(1−ρ) and µ = c+ρ−m+mρ

(m+1)(1−ρ) 6= −1. Since hi(z) ∈ P, i = 1, 2; we use

Lemma 1.6.5 to have

hi(z) +

1(m+1)(1−ρ)zh

0i(z)

hi(z) +c+ρ−m+mρ(m+1)(1−ρ)

∈ P,

for |z| < rm,c , where rm,c is given by (5.3.1) and this result is best possible. Thus, from

(5.3.2), it follows that f ∈ Rk(m, ρ) for |z| < rm,c and this completes the proof. ¥As a special case, with m = ρ = 0 and c > 0, we obtain the well-known Libra-

Livingston operator and Theorem 5.3.1 gives us r0,c = 12.

95

Chapter 6

On certain class of p-valent functions

defined by some integral operator

In this chapter, we introduce integral operators for p-valent functions in the same way

as that of Noor [71](known as Noor integral operator). For p = 1, this operator was

studied recently by Noor [68] for certain classes of analytic functions. The study of

operators in geometric function theory plays a vital role as we have seen in chapter one.

This operator is defined by convoluting the hypergeometric functions. Certain classes

of analytic functions are defined by using this operator. Some results such as inclusion

results and the properties of the generalized Bernardi integral transform are studied.

6.1 An integral operator

A function f(z) is known to be p-valent, if the equation f(z) = w has at most p zeros.

In 1935, Cartwright proved that when p is a positive integer then f(z) = w has never

more than p zeros in E for any value of w, see [29].

Let A(p) denote the class of functions f(z) of the form

f(z) = zp +∞X

n=p+1

anzn, (p ∈ N = 1, 2, 3, ...), (6.1.1)

96

which are analytic in the open unit disk E. Also A(1) = A, the usual class of analytic

functions.

For f(z) in A(p), the operator Dδ+p−1 : A(p) −→ A(p) is defined by

Dδ+p−1f(z) =zp

(1− z)δ+p∗ f(z), (δ > −p),

or equivalently, by

Dm+p−1f(z) =zp(zm−1f(z))m+p−1

(m+ p− 1)! ,

where m is any integer greater than −p. If f(z) is given by (6.1.1), then it follows that

Dδ+p−1f(z) = zp +∞X

n=p+1

(δ + n− 1)!(n− p)!(δ + p− 1)!anz

n.

The symbol Dδ+p−1 when p = 1, was introduced by Ruscheweyh [112] and the symbol

Dδ+p−1 itself was introduced by Goel and Sohi [23]. This symbol is called as the (δ+ p−

1)th order Ruscheweyh derivative. We now introduce a function (zp2F1(a, b, c; z))−1 given

by

(zp2F1(a, b, c; z)) ∗ (zp2F1(a, b, c; z))−1 =zp

(1− z)λ+p, (λ > −p),

and obtain the following linear operator

Iλ,p(a, b, c)f(z) = (zp2F1(a, b, c; z))

−1 ∗ f(z), (6.1.2)

where a, b, c are real or complex numbers other than 0,−1,−2, . . . , λ > −p, z ∈ E and

f(z) ∈ A(p).

In particular for p = 1, this operator was studied by Noor [68] recently. For b = 1

this operator reduces to the well-known Cho-Kwon-Srivastava operator Iλ,p(a, c) which

was studied by Cho et. al. [16] and for λ = 1, b = c, a = n+ p, see [92]. For a = n+ p,

b = c = 1, this operator was investigated by Liu and Noor [43](see also [60, 65, 76, 81, 82]).

97

Simple computations yield to us

Iλ,p(a, b, c)f(z) = zp +∞X

n=p+1

(c)n(λ+ p)n(a)n(b)n

anzn.

From (6.1.2), we note that

Iλ,1(a, b, c)f(z) = Iλ(a, b, c)f(z), see [68],

I0,p(a, p, a)f(z) = f(z), I1,p(a, p, a)f(z) =zf 0(z)

p.

Also, it can be easily seen that

z(Iλ,p(a, b, c)f(z))0 = (λ+ p)Iλ+1,p(a, b, c)f(z)− λIλ,p(a, b, c)f(z), (6.1.3)

and

z(Iλ,p(a+ 1, b, c)f(z))0 = aIλ,p(a, b, c)f(z)− (a− p)Iλ,p(a, b, c)f(z). (6.1.4)

From the properties of hypergeometric functions (1.6.8—1.6.12) (see also [101]), we can

write

I0,p(a, p, c)f(z) =

1Z0

µ(τ)f(τz)

τdτ ,

where

µ(τ) =Γ(a)

Γ(c)Γ(a− c)(1− τ)a−c−1F (a− c, 1− c, a− c, 1− τ),

which is the integral representation of the operator I0,p(a, p, c)f(z).

98

6.2 A class of analytic p-valent functions

Extending the concepts of bounded boundary and bounded radius rotations to p-valent

analytic functions, we define the following class of multivalent analytic functions by using

the above operator Iλ,p(a, b, c)f(z).

Definition 6.2.1

Let f(z) ∈ A(p), z ∈ E. Then f(z) ∈ Rλ,pk (a, b, c, A,B), if and only if, for k ≥ 2, A∈ C,

B ∈ [−1, 0) ,

z(z1−pIλ,p(a, b, c)f(z))0

z1−pIλ,p(a, b, c)f(z)=

µz(Iλ,p(a, b, c)f(z))

0

Iλ,p(a, b, c)f(z)− p+ 1

¶∈ Pk [A,B] .

For different values to the involved parameters in the above definition, we obtain the

following special cases:

(i). For λ = 0, a = c, b = p = 1, we have the class Rk [A,B] ⊂ Rk [1− 2ρ,−1] = Rk(ρ),

the class of functions of bounded radius rotation of order ρ.

(ii). For λ = 1, a = c, b = p = 1, we have the class Vk [A,B] ⊂ Vk [1− 2ρ,−1] = Vk(ρ),

the class of functions of bounded boundary rotation of order ρ.

6.3 Some inclusion results

Theorem 6.3.1

If f(z) ∈ Rλ+1,pk (a, b, c, A,B) with A ∈ C and B ∈ [−1, 0] satisfying either (1.6.3) or

(1.6.4), then f(z) ∈ Rλ,pk (a, b, c, 1− 2ρ1,−1) , where

ρ1 = (λ+ p)

∙2F1

µ1, 1− A

B,λ+ p+ 1;

B

B − 1

¶¸−1− (λ+ p− 1) . (6.3.1)

This result is sharp. ¤

99

Proof. Let f(z) ∈ Rλ+1,pk (a, b, c, A,B) . Setting

s(z) =(Iλ,p(a, b, c)f(z))

zp−1, (6.3.2)

logarithmic differentiation of (6.3.2) gives us

h(z) =zs0(z)

s(z)=

z (Iλ,p(a, b, c)f(z))0

Iλ,p(a, b, c)f(z)− p+ 1, (6.3.3)

where h(z) is analytic in E and h(0) = 1. Using the identity (6.1.3) in (6.3.3), we have

h(z) = (λ+ p)zIλ+1,p(a, b, c)f(z)

Iλ,p(a, b, c)f(z)− (λ+ p) + 1

= (λ+ p)

∙zIλ+1,p(a, b, c)f(z)

Iλ,p(a, b, c)f(z)− 1 + 1

λ+ p

¸.

This implies that

zIλ+1,p(a, b, c)f(z)

Iλ,p(a, b, c)f(z)=

1

λ+ p[h(z) + (λ+ p)− 1] .

Again differentiation gives us

z (Iλ+1,p(a, b, c)f(z))0

Iλ+1,p(a, b, c)f(z)= h(z) +

zh0(z)

h(z) + λ+ p− 1 .

Since f(z) ∈ Rλ+1,pk (a, b, c, A,B) , it follows that

µh(z) +

zh0(z)

h(z) + λ+ p− 1

¶∈ Pk [A,B] . (6.3.4)

Define a function

ϕa,b(z) =1

1 + b

z

(1− z)a+

b

1 + b

z

(1− z)a+1,

where a = 1, b = λ+p−1. Using the convolution technique [68, 79] together with (1.6.2),

100

it follows that

ϕa,b(z)

z∗ h(z) =

µk

4+1

2

¶ ∙ϕa,b(z)

z∗ h1(z)

¸−µk

4− 12

¶ ∙ϕa,b(z)

z∗ h2(z)

¸.

This implies that

h(z) +azp0(z)

p(z) + b=

µk

4+1

2

¶ ∙h1(z) +

azh01(z)

h1(z) + b

¸−µk

4− 12

¶ ∙h2(z) +

azh02(z)

h2(z) + b

¸.

(6.3.5)

Therefore from (6.3.4), we have

µhi(z) +

azh0i(z)

hi(z) + b

¶∈ P [A,B] , i = 1, 2.

Hence, from Lemma 1.6.6, we find that

hi(z) ≺1

g(z)− (λ+ p− 1) = q(z) ≺ 1 +Az

1 +Bz

and consequently h(z) ∈ Rλ,pk (a, b, c, 1− 2ρ1,−1) , where ρ1 is given by (6.3.1). The

sharpeness of the estimates can easily be obtained from the best dominance of the function

q(z). ¥Putting A = 1− 2ρ, B = −1, λ = 0, a = c, b = p, we have the following.

Corollary 6.3.1

For 0 ≤ ρ < p, k ≥ 2

V pk (ρ) ⊂ Rp

k(ρ1),

where

ρ1 = ρ1(ρ, p) = p

∙2F1

µ1, 2(1− ρ), p+ 1;

1

2

¶¸−1− p+ 1. ¤

Putting p = 1, in the above corollary, we obtain a result which we have published in [86].

We include the details for this result for the sake of completeness. This also provides an

101

independent proof of the following.

Corollary 6.3.2

For 0 ≤ ρ < 1, k ≥ 2

Vk(ρ) ⊂ Rk(ρ1),

where

ρ1 = ρ1(ρ) =

⎧⎨⎩4ρ(1−2ρ)4−22ρ+1 , ρ 6= 1

2,

12 ln 2

, ρ = 12.

(6.3.6)

This result is sharp. ¤Proof. Since f(z) ∈ Vk(ρ), we use Lemma 1.5.1 together with (1.5.3) to obtain

1 +zf 00 (z)

f 0 (z)=

µk

4+1

2

¶zs01(z)

s1(z)−µk

4− 12

¶zs02(z)

s2(z)

=

µk

4+1

2

¶(zf 01(z))

0

f 01(z)−µk

4− 12

¶(zf 02(z))

0

f 02(z),

where si(z) ∈ S∗ (ρ) and fi(z) ∈ C (ρ), i = 1, 2.

Now, from (1.5.9), we have

zf 0 (z)

f (z)=

µk

4+1

2

¶ zhg1(z)z

i1−ρzR0

hg1(φ)φ

i1−ρdφ

−µk

4− 12

¶ zhg2(z)z

i1−ρzR0

hg2(φ)φ

i1−ρdφ

.

102

That is

zf 0 (z)

f (z)=

µk

4+1

2

¶⎡⎣ zZ0

∙z

φ

¸1−ρ ∙g1 (φ)

g1 (z)

¸1−ρdφ

z

⎤⎦−1

−µk

4− 12

¶⎡⎣ zZ0

∙z

φ

¸1−ρ ∙g2 (φ)

g2 (z)

¸1−ρdφ

z

⎤⎦−1 , (6.3.7)

where the integration is taken along the straight line segment [0, z], z ∈ E.

Writingzf 0 (z)

f (z)=

µk

4+1

2

¶h1(z)−

µk

4+1

2

¶h2(z),

and using (6.3.7) , we have

hi(z) =

⎡⎣ zZ0

∙z

φ

¸1−ρ ∙gi (φ)

gi (z)

¸1−ρdφ

z

⎤⎦−1 ,where hi(0) = 1 and hence by [91], we have¯

hi(z)−1 + r2

1− r2

¯≤ 2r

1− r2, |z| = r, z ∈ E.

Therefore

minfi∈C(ρ)

min|z|=r

Re [hi(z)] = minfi∈C(ρ)

min|z|=r

|hi(z)| . (6.3.8)

Let z = reiθ and φ = R eiθ, 0 < R < r < 1. For fixed z and φ, we have from (1.5.8)

¯gi (φ)

gi (z)

¯≤ R

r

µ1 + r

1 +R

¶2. (6.3.9)

103

Now, using (6.3.9), we have, for a fixed z ∈ E, |z| = r,¯¯

zZ0

∙z

φ

¸1−ρ ∙gi (φ)

gi (z)

¸1−ρdφ

z

¯¯ ≤

rZ0

µ1 + r

1 +R

¶2(1−ρ)dR

r.

Let

T (r) =

rZ0

µ1 + r

1 +R

¶2(1−ρ)dR

r,

with R = rt, 0 < t < 1, we have

T (r) =

1Z0

µ1 + r

1 + rt

¶2(1−ρ)dt,

and differentiating it, we have

T 0 (r) = 2 (1− ρ)

1Z0

(1− t)

(1 + rt)2

µ1 + r

1 + rt

¶(1−2ρ)dt > 0,

and therefore T (r) is a monotone increasing function of r and hence

max0≤ r ≤1

T (r) = T (1) = 22(1−ρ)1Z0

dt

(1 + t)2(1−ρ)

=

⎧⎪⎨⎪⎩(2−4(1−ρ))(2ρ−1) , if ρ 6= 1

2,

2 ln 2, if ρ = 12.

(6.3.10)

By letting

ρ1(ρ) = min

⎡⎣¯¯zZ0

∙z

φ

¸1−ρ ∙gi (φ)

gi (z)

¸1−ρdφ

z

¯¯⎤⎦−1 , z ∈ E, (6.3.11)

for all gi (z) ∈ S∗, we obtain the required result by using (6.3.8), (6.3.10) and (6.3.11) .

104

For sharpness, consider the function f0(z) in Vk (ρ) given by

(zf 00(z))0

f 00(z)=

µk

4+1

2

¶µ1− (1− 2ρ) z

1 + z

¶−µk

4− 12

¶µ1 + (1− 2ρ) z

1− z

¶.

It is easy to check that f0(z) ∈ Rk(ρ1), where ρ1 is the exact value given by (6.3.6) . ¥For a function f ∈ A(p) given by (6.1.1), Goel and Sohi [23] introduced and studied

the integral operator Lc,p defined by

Lc,pf(z) =c+ p

zc

zZ0

tc−1f(t)dt (6.3.12)

= zp +∞X

n=p+1

c+ p

c+ p+ nanz

n, (p ∈ N, c > −p), z ∈ E.

The operator Lc,1 = Lc defined by (1.7.1) was introduced by Bernardi [7]. In particular,

the operator L1 was studied by Libera [40] and Livingston [44]. The following theorem

is an interesting application of the above results.

Theorem 6.3.2.

If f(z) ∈ Rλ,pk (a, b, c, A,B) with A ∈ C and B ∈ [−1, 0] satisfying either (1.6.3) or

(1.6.4),then Lc,pf(z) ∈ Rλ,pk (a, b, c, 1− 2ρ1,−1) , where

ρ1 = (c+ p)

∙2F1

µ1, 1− A

B, c+ p+ 1;

B

B − 1

¶¸−1− (c+ p− 1) .

This result is sharp. ¤Proof. From (6.1.2) and (6.3.12), it can easily be seen that

z(Iλ,p(a, b, c)Lc,pf(z))0 = (c+ p)Iλ,p(a, b, c)f(z)− cIλ,p(a, b, c)Lc,pf(z). (6.3.13)

Setting

s(z) =Iλ,p(a, b, c)Lc,pf(z)

zp−1, (6.3.14)

105

and differentiating (6.3.14), we have

h(z) =zs0(z)

s(z)=

z(Iλ,p(a, b, c)Lc,pf(z))0

Iλ,p(a, b, c)Lc,pf(z)− p+ 1, (6.3.15)

where h(z) is analytic in E and h(0) = 1. Using the identity (6.3.13) in (6.3.15), we have

h(z) = (c+ p)(Iλ,p(a, b, c)f(z)

Iλ,p(a, b, c)Lc,pf(z)− c− p+ 1.

It implies thatIλ,p(a, b, c)Lc,pf(z)

Iλ,p(a, b, c)f(z)=

c+ p

h(z) + c+ p− 1 . (6.3.16)

Differentiating (6.3.16) and using (6.3.15), we have

z(Iλ,p(a, b, c)f(z))0

Iλ,p(a, b, c)f(z)= h(z) +

zh0(z)

h(z) + c+ p− 1 + p− 1.

Since f(z) ∈ Rλ,pk (a, b, c, A,B) , it follows that

∙h(z) +

zh0(z)

h(z) + c+ p− 1 + p− 1¸∈ Pk [A,B] , (z ∈ E).

Using (1.6.2) and convolution technique [68, 79], we have

∙hi(z) +

zh0i(z)

hi(z) + c+ p− 1 + p− 1¸∈ P [A,B] , i = 1, 2.

Hence, by using Lemma 1.6.6 , we find that

hi(z) ≺1

g(z)− (c+ p− 1) = q(z) ≺ 1 +Az

1 +Bz, z ∈ E, i = 1, 2

and consequently h(z) ∈ Pk [A,B] and the function q(z) is best dominant. The sharpness

can be viewed from the best dominance of the function q(z). ¥For p = 1, A = 1 − 2ρ, B= −1, a = c, b = 1, λ = 0 in Theorem 6.3.2, we have the

following interesting result.

106

Corollary 6.3.3

For 0 ≤ ρ < 1, if f(z) ∈ Rk(ρ), then integral operator F (z) defined by (1.7.1) belongs to

the class Rk(ρ1), where

ρ1 = ρ1(ρ) =c+ 1

2F1(1, 2− 2ρ, c+ 2, 12)− c.

This result is sharp. ¤Proof. The proof follows at once when we proceed as in the above theorem and apply

Lemma 1.6.6. The sharpness of the estimates follows from the best dominance of q(z).

¥

107

ConclusionThis research is mainly concerned with the analytic functions defined in open unit disk E.

In this thesis, certain classes of analytic functions, such as κ−UKη(λ, α), κ−UQη(λ, α),eRk(γ1, γ2, β), eNk(η, ρ, β), Rk(m, ρ) and Rλ,pk (a, b, c, A,B), were introduced and their var-

ious interesting properties were investigated. These classes generalized the concepts of

uniformly close-to-convex and quasi-convex, strongly close-to-convex functions, bounded

turning, bounded boundary rotations and bounded radius rotations. The techniques of

convolution and differential subordination were employed to investigate certain problems

such as inclusion results, radius problem, arc length problem, coefficients and growth of

Hankel determinant problem and several other interesting properties of the above men-

tioned classes. We observed that some well known results are reduced as special cases

from our main results signifying the work presented in this thesis.

This work will motivate researchers working in this field to find many new applications

in their related areas. We hope, this will open new directions of research in this field.

108

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