Certain Subclasses of Analytic Functions Associated with Conic Domains

By

Sarfraz Nawaz CIIT/SP09–PMT–015/ISB

PhD Thesis

In Mathematics

COMSATS Institute of Information Technology

Islamabad, Pakistan Spring, 2012

ii 

 

COMSATS Institute of Information Technology

Certain Subclasses of Analytic Functions

Associated with Conic Domains

A Thesis Presented to

COMSATS Institute of Information Technology, Islamabad

In partial fulfillment

of the requirement for the degree of

PhD Mathematics

By

Sarfraz Nawaz

CIIT/SP09–PMT–015/ISB

Spring, 2012

iii 

 

Certain Subclasses of Analytic Functions Associated with Conic Domains

A Post Graduate thesis submitted to the Department of Mathematics as partial fulfillment of the requirement for the award of Degree of PhD Mathematics.

Name Registration Number

Sarfraz Nawaz CIIT/SP09–PMT–015/ISB

Supervisor Dr. Khalida Inayat Noor Professor, Department of Mathematics Islamabad Campus. COMSATS Institute of Information Technology (CIIT) Islamabad Campus. July, 2012

iv 

 

Final Approval

This thesis titled

Certain Subclasses of Analytic Functions

Associated with Conic Domains By

Sarfraz Nawaz CIIT/SP09 –PMT– 015/ISB

Has been approved For the COMSATS Institute of Information Technology, Islamabad.

External Examiner 1 : __________________________ Prof. Dr. Ghulam Shabbir Professor GIKI, Topi KPK, Pakistan

External Examiner 2 : __________________________

Prof. Dr. Noor Ahmed Sheikh Professor University of Sindh, Jamshoro, Pakistan

Supervisor : __________________________

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

HoD : __________________________ Dr. Moiz ud Din Khan Associate Professor

CIIT, Islamabad, Pakistan

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

v 

 

Declaration

I, Sarfraz Nawaz, CIIT/SP09–PMT–015/ISB hereby declare that I have produced the

work presented in this thesis, during the scheduled period of study. I also declare that I

have not taken any material from any source except referred to wherever due that amount

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

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

the HEC.

Date:

Signature of student:

Sarfraz Nawaz

CIIT/SP09–PMT– 015/ISB

vi 

 

Certificate It is certified that Sarfraz Nawaz CIIT/SP09-PMT-015/ISB has carried out all the work

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

COMSATS Institute of Information Technology, Islamabad and the work fulfills the

requirement for award of PhD degree.

Date: ________________

Supervisor:

___________________________ Prof. Dr. Khalida Inayat Noor Professor

Head of Department:

__________________

Dr. Moiz ud Din Khan

Associate Professor

Mathematics Department

vii 

 

DEDICATION

This work is dedicated

To

My Father Muhammad Nawaz Malik

Who

Always has Faith in me and his Faith and Love made me what I am today.

viii 

 

ACKNOWLEDGEMENTS

All thanks and acknowledgments to Almighty ALLAH, the Most Merciful and

Compassionate, the Most Gracious and Beneficent, Who is the sole source of knowledge

and wisdom, Who has endowed human beings with the faculty of reasoning and pursuing

knowledge and Who through his Kindness, Graciousness and Countless Blessings

enabled me to pursue higher ideals in life.

I wish to assert my sincere, humble and deep gratitude to my Respected Supervisor

Prof. Dr. Khalida Inayat Noor, Department of Mathematics, CIIT, Islamabad, for her

constant and scholarly guidance, enriching and enlightening suggestions, precious and

insightful advice and her kind and sympathetic demeanor, which not only enabled me to

carry out this research work and bring my thesis into final shape but also helped me to

build a strong and viable foundation for the future research ventures. Besides providing

wonderful insight into research areas, she has left deep and lasting imprints on my

thinking and personality.

I also humbly acknowledge insightful and inspiring guidance of Respected

Prof. Dr. Muhammad Aslam Noor, Department of Mathematics, CIIT, Islamabad, for

his scholarly, perceptive and insightful guidance during my course work.

I also humbly acknowledge and appreciate the role of Honorable Rector

Dr. S.M. Junaid Zaidi, CIIT, Pakistan, the Dean, Faculty of Sciences and the

Head, Department of Mathematics, CIIT, Islamabad, in providing a truly international

standard research environment, state of the art research facilities and a student friendly

atmosphere.

Words always fail to me to acknowledge my parents who have always been guiding stars

and a constant source of inspiration and encouragement for me. I also acknowledge the

moral and financial support of my family members who have always backed me up and

motivated me to achieve my goals.

ix 

 

I also acknowledge and appreciate the open and warm hearted support and helpful

advices of my fellow Mohsan and seniors Asif, Saira, Zakir who always spared

precious hours for me inspite of their own hectic schedules.

Last but not the least, I also express my deep gratitude to Higher Education

Commission of Pakistan for providing me the much needed financial support in the

form of Indigenous Fellowship Program as well as the access to the latest research.

Sarfraz Nawaz

CIIT/SP09–PMT– 015/ISB

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ABSTRACT

Certain Subclasses of Analytic Functions Associated

with Conic Domains

Geometric Function Theory is the branch of Complex Analysis which deals with the

study of geometric properties of analytic functions. While studying the geometric

properties of analytic functions, we are mainly concerned with the geometry of image

domains of analytic functions. On the basis of shape and other properties of image

domain, analytic functions are categorized into many classes and then into subclasses.

Only a few geometrical structures have been introduced as image domain in which conics

is of great importance.

The main focus of this study is to develop conic domains and to introduce some new

geometrical structures as image domains. Our aim is to develop and refine already known

conic domains and also to introduce certain new generalized domains and their associated

functions. Also we deal with generalized circular domain and introduce certain new

classes of analytic functions representing conic and circular domains simultaneously

which is the main motivation of this work.

TABLE OF CONTENTS

1 Introduction 8

1.1 Historical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 Geometric Function Theory in today�s sciences . . . . . . . . . . . . . . . 11

1.3 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Preliminary Concepts and De�nitions 17

2.1 Analytic and univalent functions . . . . . . . . . . . . . . . . . . . . . . . 18

2.1.1 Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1.2 Univalent Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 The class P of functions with positive real part . . . . . . . . . . . . . . 21

2.3 Some subclasses of univalent functions . . . . . . . . . . . . . . . . . . . 23

2.4 Certain subclasses of analytic functions of order beta . . . . . . . . . . . 26

2.5 Subordination and di¤erential subordination . . . . . . . . . . . . . . . . 27

2.6 Conic domains and associated functions . . . . . . . . . . . . . . . . . . . 28

2.6.1 Hyperbolic, parabolic and elliptic domains . . . . . . . . . . . . . 28

2.6.2 Circular domains . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.7 The class Vm of functions with bounded boundary rotation and related

classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.7.1 The class Vm of functions with bounded boundary rotation . . . . 36

2.7.2 The class Rm of functions with bounded radius rotation . . . . . . 37

2.7.3 The class Pm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.8 Ruscheweyh derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

1

2.9 Hypergeometric functions . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.9.1 Con�uent hypergeometric function . . . . . . . . . . . . . . . . . 40

2.9.2 Gaussian hypergeometric function . . . . . . . . . . . . . . . . . . 41

2.10 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3 On Generalized �-Convex Functions Associated with Conic Domain 44

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4 On a New Class of Analytic Functions Associated with Conic Domain 61

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5 On Bounded Boundary and Bounded Radius Rotation Related with

Janowski Function 78

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.2 Preliminary Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6 OnGeneralization of a Class of Analytic Functions De�ned by Ruscheweyh

Derivative 93

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.2 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7 On Janowski Functions Associated with Conic Domains 108

2

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

7.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

8 Conclusion 129

9 References 131

3

4

LIST OF FIGURES

Fig 2.1 Convex and starlike domains ...………………………………………24

Fig 2.2 Boundaries of conic regions defined by kΩ ...……………………….30

Fig 4.1 View of 1.2 , 0.5Ω ………………………………………………………65

Fig 4.2 Contraction of kΩ ……………………………………………………66

Fig 4.3 Magnification of kΩ …………………………………………………66

Fig 7.1 Boundary of domain [ ]0.5, 0.5kΩ − ………………………..………111

Fig 7.2 Boundary of domain [ ]0.8,0.2kΩ …………………..…...…………111

Fig 7.3 View of [ ],k A BΩ when 1, 1A B→ →− ……….…….….………..112

Fig 7.4 Close view of Fig. 7.3 ……………………………….…………..….112

5

LIST OF ABBREVIATIONS

Set of complex numbers

E Open unit disk

A The class of normalized analytic functions

S Class of univalent functions

C Class of convex univalent functions *S Class of starlike univalent functions

( )k z Koebe function

P Class of analytic functions with positive real parts

( )0L z Möbius function

Mα Class of α -convex functions

( )P β Class of analytic functions with real part greater than beta

( )C β Class of convex univalent functions of order beta

( )*S β Class of starlike univalent functions of order beta

≺ Subordination symbol

Δ Family of Schwarz functions

UCV Class of uniformly convex functions

UST Class of uniformly starlike functions

ST Class of corresponding starlike functions

Ω Parabolic domain

kΩ Conic domain

k UCV− Class of k-uniformly convex functions

6

k ST− Class of k-starlike functions

( ),G k β , ,k βΩ Conic domain of order beta

( ),KD k β Class of k-uniformly convex functions of order beta

( ),SD k β Class of k-starlike functions of order beta

[ ],P A B Class of Janowski functions

[ ],A BΩ Circular domain

[ ],C A B Janowski convex functions

[ ]* ,S A B Janowski starlike functions

[ ], ,A B βΩ Circular domain of order beta

[ ], ,P A B β Class of Janowski functions of order beta

mV Class of functions with bounded boundary rotation

mR Class of functions with bounded radius rotation

mP Class of functions with bounded turnings

f g∗ Convolution of f and g

Dδ Ruscheweyh derivative of order δ

( ). n Pochhammer symbol

( )1 1 , ;F a c z Confluent hypergeometric function

( )2 1 , , ;F a b c z Gaussian hypergeometric function

( ),UM k α Class of α -convex functions associated with conic domain

( ), ,k UB α β γ− Class of generalizedα -convex functions associated with

conic domain

( ),k a bΩ Generalized conic domain

7

( ),k UCV a b− Class of k-uniformly convex functions of type ( ),a b

( ),k ST a b− Class of k-starlike functions of type ( ),a b

[ ], ,mV A B β Class of Janowski functions with bounded boundary

rotation and order beta

[ ], ,mR A B β Class of Janowski functions with bounded radius rotation

and order beta

[ ],k A BΩ Modified conic domain

[ ],k UCV A B− Class of k-uniformly Janowski convex functions

[ ],k ST A B− Class of k-starlike Janowski functions

Chapter 1

Introduction

8

1.1 Historical background

Geometric Function Theory is the branch of complex analysis which deals with the geo-

metric properties of analytic functions. This branch of mathematics is not as old as

the other branches are. We know that there are mappings in complex valued functions

f (z) ; z 2 D � C; in which the image domains f (D) are of major interest. The analytic

functions got more attraction in 1851, when Bernard Riemann allowed to replace any

arbitrary domain D with the open unit disk E = fz : jzj < 1g, by giving the famous

Riemann Mapping Theorem. This theorem gave rise to the birth of fascinating area of

mathematics, called Geometric Function Theory. Riemann made signi�cant contribution

to develop this area of mathematics.

In 1907, Koebe [24] initiated the theory of univalent functions by discovering functions

which are both analytic and univalent in open unit disk E: The set of such functions f

which are analytic and univalent in open unit disk E and are normalized by the conditions

f (0) = 0; f 0 (0) = 1 was then called the class S: This class S became the core ingredient

of advance research in this �eld. Many subclasses of class S of univalent functions were

de�ned on the basis of geometry of image domains, from which the classes C and S�

of convex and starlike univalent functions respectively have been of major interest. The

class P of functions with positive real part played an important role in setting the criteria

for convexity and starlikeness of univalent functions. In 1913, Study [105] used this class

P in setting the criteria for convexity of univalent functions. In 1915, Alexander [7]

established a beautiful relation between these classes of starlike and convex functions.

After Koebe, Bieberbach[24] in 1916 proved the second coe¢ cient estimate of func-

tions of S. He made remarkable contribution to this �eld by giving a famous open

problem regarding the coe¢ cient estimates of functions of S and named this problem as

Bieberbach�s conjecture. This problem remained open as a challenge for a wide number

of mathematicians working in this area and was �nally settled by de-Branges [19] in

1984. Bieberbach�s conjecture lasted as an ambitious problem of the �eld and got the at-

traction of many researchers. Due to complications in proving the famous Bieberbach�s

9

conjecture, researchers used to prove it for particular types of univalent functions. In

1921, Nevanlinna [55] proved the coe¢ cient estimate of starlike functions and gave the

criteria for starlikeness of univalent functions, see [24]. After that, a chain of researchers

contributed a lot to many developments and innovations in these two basic subclasses of

univalent functions and applied several di¤erential and integral operators, see for example

[20, 24, 27, 38, 53, 90, 91, 98, 103].

In 1917, Löwner [50] introduced the concept of functions with bounded boundary

rotation which was developed in more re�ned and systematic way by Paatero [84]. He

denoted the class of function with bounded boundary rotation by Vm; m � 2: Tammi

[106] introduced the class Rm; m � 2 of functions with bounded radius rotation in

1952. These concepts of bounded boundary and bounded radius rotation opened a new

direction of research and drew much attention of scholars. Several authors like Kirwan

[37], Brannan [11, 12], Pinchuk [87] and Noor [61, 64, 69, 76, 77, 78] wrote remarkable

research articles and played signi�cant role in developing this area of Geometric Functions

Theory. In 1971, Pinchuk [87] introduced the class Pm; m � 2 by using the concept of

functions with bounded boundary rotations and gave the criterion for functions to be

from Vm or Rm:

As we have discussed in the beginning that while studying the analytic functions f;

their image domain f (E) is of great importance. On the basis of the shapes of image

domains, various classes have been de�ned and studied. In 1973, by using the concept of

subordination which was given by Lindelöf [41], Janowski [30] introduced the concept of

circular domain [A;B] and de�ned the class P [A;B] of functions which map the open

unit disk onto the circular regions, called Janowski functions. A number of renowned

researchers studied these functions and several new classes of analytic functions associated

with circular domains have been introduced and studied, see for example [26, 47, 67, 68].

In 1991, Goodman [22, 23] initiated the idea of conic domains by introducing the classes

of uniformly convex and uniformly starlike functions. Later, Rønning [95], Ma and Minda

[51] independently found a most suitable form of Goodman criteria of these classes and

10

re�ned the parabolic domain. With the analogy of Alexander relation, Rønning [97]

introduced and studied the class of starlike functions corresponding to uniformly convex

functions, we will discuss these developments in section 1.3. Finally, in 1999, Kanas and

Wisniowska [36, 35] generalized the parabolic domain and introduced the general conic

domain k; k � 0 which represents all the three conic structures(hyperbola, parabola

and ellipse). They contributed remarkably in the study of conic domains. Furthermore, a

large number of renowned mathematicians like Sirivastava, Noor, Acu, Owa etc. studied

these conic domains, see [1, 2, 3, 34, 82, 104].

Geometric Function Theory is a wide research area. In above, we took a glimpse of

the developments which lead us to the theme of this thesis and to keep into the essence,

we omit the rest of developments in this �eld.

1.2 Geometric Function Theory in today�s sciences

Geometric Function Theory has recently found many applications in various �elds of ap-

plied sciences such as nonlinear integrable system theory, �uid dynamics, modern math-

ematical physics and the theory of partial di¤erential equations. Conformal mapping is

used to solve eigen value problem in plane and that method is called conformal transplan-

tation, see [38]. The theory of compact Riemann surfaces is widely used in constructing

�nite-gap solutions to nonlinear integrable systems, see [10] To compute the uniform po-

tential �ow around an assembly of circular obstacles is a basic and fundamental problem

in �uid mechanics and it is also a prototypical example of a physical problem involving a

multiply connected domains. The Schottky-Klein prime function is a special function of

complex variable which proves to be very useful in solving a range of problems in Geo-

metric Function Theory. Crowdy [17] made a comprehensive survey on the applications

of Geometric Functions Theory and used the Schottky-Klein prime function in prob-

lems involving multiply connected domain. For more applications of Geometric Function

Theory, we refer to [17, 18, 25, 107, 108].

11

1.3 Preface

As it is evident from the title "Certain Subclasses of Analytic Functions Asso-

ciated with Conic Domains" of this thesis, we are mainly focusing on the study of

conic domains including the circular domains, because somehow circle is a special case of

ellipse. But in the study of image domains, we see that circular domains are de�ned in a

distinctive way and are not deduced from general conic domains as special cases, so we

study them in separate manner. Our aim is to develop and re�ne these already known

conic domains and also to introduce certain new generalized domains and their associ-

ated functions. Also we deal with generalized circular domain and introduce certain new

classes of analytic functions representing conic and circular domains simultaneously.

From the above sequential historical overview of this �eld, we �nd that Goodman

[22, 23] in 1991 initiated the concept of conic domain but unintentionally. It is well

known that for any convex function f (z) ; not only f (E) but also the images of all

circles lying in E centered at origin are convex arcs. Pinchuk raised a question that

whether this property remains valid for circles centered at other points or not. Goodman

[22] replied in negative and introduced the class of functions which obey this property.

He named such functions as uniformly convex functions and denoted the class of such

functions by UCV: He also introduced the class UST of uniformly starlike functions

which map the circles lying in E centered at other points onto star shaped arcs, see [23].

He not only just de�ned these classes but also gave the analytic conditions for uniform

convexity and uniform starlikeness. By using these analytic conditions, these classes can

be de�ned as

UCV =

�f 2 A : Re

�1 + (z � �)

f 00 (z)

f 0 (z)

�> 0; z; � 2 E

�;

UST =

�f 2 A : Re

�(z � �) f 0 (z)

f (z)� f (�)

�> 0; z; � 2 E

�:

These classes coincide with the classes C and S� respectively when we take circles centered

12

at origin (that is � = 0). The famous Alexander�s relation stating the fact that f 2 C

if and only if zf 0 2 S� is fails to hold between UCV and UST: This fact is evident

from above de�nitions and can also be illustrated by two counter examples, given in [22].

Due to this reason, the class UST became unpopular but Rønning [97] �lled this gap by

introducing the class

ST = ff 2 A : f (z) = zg0 (z) ; g 2 UCV g

which is associated with the class UCV by Alexander type relation. He further proved

in [97] that neither UST ST nor ST UST: Later, again Rønning [95], and Ma

and Minda [51] independently gave the most suitable one variable characterization of the

class UCV and de�ned it as

UCV =

�f 2 A : Re

�1 +

zf 00 (z)

f 0 (z)

�>

����zf 00 (z)f 0 (z)

���� ; z 2 E� :Similarly, the class ST took the following form.

ST =

�f 2 A : Re

�zf 0 (z)

f (z)

�>

����zf 0 (z)f (z)� 1���� ; z 2 E� :

This characterization gave birth to the �rst conic (parabolic) domain

= fw : Rew > jw � 1jg :

This domain was then generalized by kanas and Wisniowska [36, 35] and introduced the

domain

k = fw : Rew > k jw � 1j ; k � 0g :

This conic domain represents the right half plane for k = 0; hyperbolic regions when

0 < k < 1; parabolic region for k = 1 and elliptic regions when k > 1: They also

introduced the class k�UCV of k-uniformly convex functions and k�ST of corresponding

13

k-starlike functions.

This thesis is aimed with the study of these classes associated with conic domains

and to introduce certain new classes of analytic functions associated with conic domains.

The chapter wise overview is as under.

Chapter 2 consists of some preliminary concepts and de�nitions. It starts with

the concept of analytic functions and normalized univalent functions in open unit disk

E and covers the basic subclasses of univalent functions. The class P of functions with

positive real parts is discussed in some detail and its generalization with regard to order �

(0 � � < 1) is discussed. The concept of subordination followed by a short survey of conic

domains is included. Also a detailed introduction of circular domains given by Janowski

[30] is provided. The functions with bounded boundary and bounded radius rotation are

reviewed. A well-known linear operator named as the Ruscheweyh di¤erential operator is

given which will be used in one of our upcoming chapters. At the end, some preliminary

results are included which will be utilized in subsequent chapters. It is important to

mention that no new de�nition or result is included in this chapter and all the contents

are known and properly referred.

The class UM (�; k) of �-convex functions associated with conic domains is intro-

duced by Kanas [31] and this class gives k-uniformly convex and corresponding k-starlike

functions as special cases. Also the class M� of �- convex functions is obtained as the

special case of UM (�; k) as UM (�; 0) � M�: In chapter 3, we generalize the class

UM (�; k) and introduce the generalized class k � UB (�; �; ) : This class gives several

known classes as special cases. Many interesting properties of this class are investigated

here and many known results are deduced from our main results as special cases. This

work has been published in "Mathematical and Computer Modelling, Vol 55; 2012; Pages

844� 852", see [74].

As mentioned earlier, the conic domain k; k � 0 was introduced and comprehen-

sively studied by Kanas and Wisniowska [36, 35]. These conic regions were restricted

to have �xed vertices and lengths of latus rectum. These conic domains could neither

14

be contracted nor magni�ed. We have recently removed this de�ciency of conic domain

k; k � 0 by introducing a generalized conic domain k (a; b) ; k � 0: This generalized

conic domain gives contraction as well as magni�cation of the conic domain k; k � 0: A

number of conic regions of any size can be obtained from this generalized conic domain

k (a; b) ; k � 0 by assigning suitable values to parameters a and b: In chapter 4, we

have given a detailed overview of generalized conic domain k (a; b) ; k � 0 and discussed

all of its aspects. Di¤erent views of this generalized conic domain for speci�c values of pa-

rameters are shown graphically for better understanding of the behaviour of this domain.

The class k�P (a; b) of functions which map the open unit disk E onto these generalized

conic regions is de�ned and some of its properties are discussed. Related to the class

k� P (a; b) ; two more classes k�UCV (a; b) and k� ST (a; b) are introduced and some

results concerning these classes are investigated. All contents of this chapter have been

published in a well reputed journal "Computers and Mathematics with Applications, Vol

62; 2011; Pages 367� 375", see [70].

In chapter 5, we have studied the class P [A;B; �] of generalized Janowski functions

of order � along with the functions with bounded boundary and bounded radius rotation.

The order of a function from the class Vm[A;B; �] of Janowski functions with bounded

boundary rotation to be from Rm[A;B; �] of Janowski functions with bounded radius

rotation is of major interest and some of its applications are also the part of our discussion.

Our main results present an advancement of already known results. It is to be mentioned

here that this work has also been published in "World Applied Sciences Journal, Vol

12(6) 2011, Pages 895� 902", see [79].

In 1994, Latha and Nanjunda [39] introduced the class Vm (�; b; �) of analytic func-

tions by using the Ruscheweyh derivative. In chapter 6, we have generalized this class

Vm (�; b; �) with the concept of Janowski functions and introduced the class V�m [A;B; �; b] :

This generalized class contains many known classes. The coe¢ cient bound, inclusion re-

sult and a radius problem are investigated here. Several known results are also deducted

from our main results as special cases by assigning particular values to di¤erent parame-

15

ters.

In the study of analytic functions, geometry of image domain is no doubt of great

importance. Analytic functions are classi�ed into many classes and then into subclasses

depending upon the shape of image domain and other geometrical properties. It has

been a matter of discussion that there always exist analytic functions with di¤erent

geometrical structures as their image domains. In chapter 7, we have introduced a

few geometrical structures of oval and petal type shape as image domain which are

new in nature and de�ned the classes of functions which give these types of mappings.

The concepts of Janowski functions and conic domains are combined together to de�ne

a new domain k [A;B] which represents the oval and petal type regions. Di¤erent

graphical views of this new domain for speci�c values of parameters are shown in order

to have better understanding of the behaviour of this domain k [A;B] : This domain

gives both conic and circular domains as special cases which is the main motivation of

this chapter. Moreover, two new classes k�UCV [A;B] of k-uniformly Janowski convex

and k � ST [A;B] of k- Janowski starlike functions are introduced and our main results

are based on their properties. A number of already known classes of analytic functions

can easily be obtained from our new classes as special cases. The class SD (k; �) is also a

special case of our new class k�ST [A;B] : The coe¢ cient bound for the class SD (k; �),

proved by Owa et al. [83] is improved, that is, the coe¢ cient bound for class SD (k; �)

obtained from our main results as special case gives much better results as compared

to that one, proved by Owa et al. [83]. It is worthy to be mentioned that some of the

contents of this chapter have also been published in a well reputed journal "Computers

and Mathematics with Applications, Vol 62; 2011; Pages 2209� 2217", see[73].

16

Chapter 2

Preliminary Concepts and De�nitions

17

In this chapter, we will give a brief introduction of elementary concepts in order to have

a better understanding of the work presented in this thesis. It will start with the concept

of analytic and univalent functions. The class S of normalized univalent functions along

with its basic subclasses will be discussed. we will also give basic de�nition and properties

of functions with positive real part and discuss the functions with real part greater than

� (0 � � < 1). A short survey of conic domains with various interesting properties

will also be given. The concept of circular domains given by Janowski functions will be

included in detail. At the end, there will be some preliminary known results to be used

in henceforth coming chapters.

2.1 Analytic and univalent functions

In this �rst section, we brie�y discuss the class A of normalized analytic functions de�ned

in the open unit disk E = fz : jzj < 1g and the class S of normalized univalent functions.

Some properties of the class S are also discussed.

2.1.1 Analytic Functions

Analytic functions play a vital role in Geometric Function Theory. They are de�ned as

follows.

De�nition 2.1.1 A complex valued function w = f (z) of complex variable z is said to

be analytic at the point z0 in the domain D; if it is di¤erentiable at z0 and at every point

in the neighbourhood of z0: The function w = f (z) is analytic in D; if it has derivative

at each point of D:

The functions which are analytic in the whole complex plane are called entire func-

tions. The functions exp (z) ; sin z; cos z and the polynomial function are entire in nature.

For more details, see [20, 24].

18

In Geometric Function Theory, we are always concerned with a domain D: An as-

sumption of replacing an arbitrary domain D with an open unit disk E = fz : jzj < 1g

can be made due to the following Riemann Mapping Theorem [20, 24].

Theorem 2.1.1 (Riemann Mapping Theorem) For every simply connected domain

D � C with at least two boundary points, there exists a unique analytic function which

maps D onto the open unit disk E = fz : jzj < 1g :

Now we de�ne the basic class A of normalized analytic functions as follows.

De�nition 2.1.2 A function f is said to be in the class A; if it is analytic in the open

unit disk E and is normalized by the conditions f (0) = 0 and f 0 (0) = 1: That is, the

class A consists of analytic functions having Taylor�s series expansion of the form

f (z) = z +1Xn=2

anzn; z 2 E: (2.1.1)

2.1.2 Univalent Functions

A single valued function f is said to be univalent in an open unit disk E if it provides a

one-to-one correspondence between the open unit disk E and the image domain f (E) :

Various other terms (like simple or schlicht) are used for this concept. The domain f (E)

of univalent function f is simple domain (a domain which is not self overlapping). More

precisely, we de�ne univalent functions as follows. For details, we refer to [20, 24].

De�nition 2.1.3 A single valued function f is said to be univalent in an open unit disk

E if for z1; z2 2 E;

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

The function f is said to be locally univalent at a point z0 2 E if it is univalent in

some neighborhood of z0:

We are mainly interested in univalent functions that are also analytic in E. Such

functions form the class S which is de�ned as follows.

19

De�nition 2.1.4 The functions analytic, univalent and normalized by the conditions

f (0) = 0 and f 0 (0) = 1 are said to form the class S: That is

S = ff 2 A : f is univalent in Eg :

The leading example of a function of class S is the Koebe function

k (z) =z

(1� z)2= z +

1Xn=2

nzn: (2.1.2)

The Koebe function maps E onto the entire plane minus the negative real axis from �14

to �1: Some other examples of functions in S are as follows.

1. f (z) = z; the identity mapping.

2. f (z) = z1�z2 ; which maps E onto the entire plane except the two half lines

12� x <

1 and �1 < x � �12:

3. f (z) = 12log�1+z1�z�; which maps E onto the horizontal strip ��

4< Im (w) < �

4:

The class S is preserved under a number of elementary transformations, see [20, 24].

Some of these transformations are given in the following theorem.

Theorem 2.1.2 [24] If f 2 S; then each of the following functions g is in S:

1. Conjugation

g (z) = f (z):

2. Rotation

g (z) = e�i�f�ei�z

�; � real.

3. Dilation

g (z) =1

tf (tz) ; 0 < t < 1:

20

4. Disk automorphism

g (z) =f�z+�1+�z

�� f (�)�

1� j�j2�f 0 (�)

; j�j < 1:

5. Omitted-value transformation

g (z) =f (z)

1� f(z)!

; where f (z) 6= !:

6. k-fold symmetry

g (z) = kpf (zk); where k is a positive integer.

In 1916, Bieberbach conjectured that if f 2 S and has the form (2:1:1), then janj �

n; n � 2; see [24]. Many attempts were made to prove it. A number of mathematicians

tried to prove it but �nally, Louis De�Branges got the honor to settle this conjecture and

now it is named as �de�Branges Theorem�, see [19].

2.2 The class P of functions with positive real part

It was observed that wherever there are so many complex valued functions whose image

domains cover the whole complex plane, there also exist functions with image domains

restricted to the open half plane. It was needed that such functions should be normalized

as it was done in the study of univalent functions. The class of such functions was named

as the class P; see [24].

De�nition 2.2.1 The class P consists of those analytic functions p which are normalized

by the condition p (0) = 1 and Re p (z) > 0; z 2 E: That is,

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

cnzn if and only if Re p (z) > 0; z 2 E: (2.2.1)

21

The most common example of functions from this class is the Möbius function

L0 (z) =1 + z

1� z: (2.2.2)

We note that

1. The class P is a convex set.

2. The function p 2 P need not be univalent. Justi�cation can be given by taking

p (z) = 1 + zn because it is in P for all n � 0 but it is not univalent for n � 2:

Herglotz [28] de�ned the function from class P in some other way by introducing their

valuable integral representation as follows.

Theorem 2.2.1 A function p is in the class P if and only if it can be expressed as

p (z) =1

2�

2�Z0

1 + ze�it

1� ze�itd� (t) ; for every z 2 E;

where � (t) is a non-decreasing real valued function such that

2�Z0

d� (t) = 2�:

In the following, we discuss the coe¢ cient estimate of functions from the class P; see

[24].

Theorem 2.2.2 Let p 2 P and be given by (2:2:1) : Then, for n � 1;

jcnj � 2:

This bound is sharp. Equality holds for L0; given by (2:2:2) :

22

In 1935, Noshiro [81] and Warschawski [109] independently gave the following beau-

tiful and simple criteria for univalence which connects the class P to the univalent func-

tions. They used the general convex domain which we will discuss in next section in some

detail.

Theorem 2.2.3 (Noshiro-Warschawski Theorem) Suppose that for some real �; we

have

Re�ei�f 0 (z)

�> 0

for all z in a convex domain D: Then f is univalent in D:

For proof, see [24].

2.3 Some subclasses of univalent functions

Analytic functions are classi�ed into certain subclasses on the basis of geometry of their

image domains. The most commonly studied classes are the classes of starlike, convex,

close-to-convex and quasi-convex functions. This classi�cation was started when the at-

tempts to prove the Bieberbach conjecture were made. Due to the complications and then

failure in proving the coe¢ cient bound for the class S of univalent functions, researchers

used to prove it for particular types of univalent functions and as a result we have many

subclasses of univalent functions which have also been studied in multivalent feature.

Now we take a little view of certain subclasses of normalized univalent functions which

are de�ned by natural geometrical conditions. We will also discuss their inter relations

and coe¢ cient bounds. For details, we refer to [20, 24].

De�nition 2.3.1 A set D in a plane is said to be starlike with respect to a point w0 2 D

if the linear segment joining w0 to every other point w 2 D lies entirely in D: In more

picturesque language, the requirement is that every point of D be �visible� from w0: A

function which maps the open unit disk E onto a domain that is starlike with respect

to the origin is called starlike function. The subclass of S consisting of all the starlike

23

functions is denoted by S�: The Koebe function de�ned by (2:1:2) is the best example of

starlike function. Example of starlike domain is shown in Figure 2.1 (ii).

De�nition 2.3.2 A set D in a plane is said to be convex if it is starlike with respect to

each of its points; that is, if the linear segment joining any two points of D lies entirely in

D: A function which maps the open unit disk E onto a convex domain is called a convex

function. The subclass of S consisting of all the convex functions is denoted by C: The

function

f (z) =z

1� z

is the leading example of the convex function. Example of convex domain is shown in

Figure 2.1 (i).

Figure 2.1: Convex and starlike domains

The following two theorems give an analytic description of starlike and convex functions.

The following theorem is due to Nevanlinna [55].

Theorem 2.3.1 A function f 2 S is said to be in the class S� of starlike univalent

functions if and only ifzf 0 (z)

f (z)2 P: (2.3.1)

The following analytic condition for convex functions is due to Study [105].

24

Theorem 2.3.2 A function f 2 S is said to be in the class C of convex univalent

functions if and only if(zf 0 (z))0

f 0 (z)2 P: (2.3.2)

It is noted that C � S� � S: In 1915, Alexander [7] built the following beautiful

relation between the classes of starlike and convex functions, for more detail, see[24].

Theorem 2.3.3 (Alexander�s Relation) Let f 2 A: Then f 2 C if and only if zf 0 2

S�:

This can be rephrased as:

If F (z) is in S�; thenzR0

F (�)�d� is in the class C:

The following two theorems give the coe¢ cient bounds for starlike and convex func-

tions, for reference, see [24].

Theorem 2.3.4 Let f 2 S� and be given by (2:1:1) : Then for z 2 E;

janj � n; n = 2; 3; 4; : : : :

This bound is sharp and equality holds for any rotation of the Koebe function de�ned by

(2:1:2) :

Theorem 2.3.5 Let f 2 C and be given by (2:1:1) : Then for z 2 E;

janj � 1; n = 2; 3; 4; : : : :

This bound is sharp and equality holds for the function f0 (z) = z1�z :

In 1969, Mocanu [54] introduced the concept of ��convexity as follows.

De�nition 2.3.3 Let � � 0 and if f 2 A; f(z)zf 0 (z) 6= 0 and

(1� �)zf 0 (z)

f (z)+ �

(zf 0 (z))0

f 0 (z)2 P;

25

then f is said to be an �-convex function. The class of such functions will be denoted by

M�:

We see that M0 � S� and M1 � C: It has been shown in [54] that M� � S�; � � 0

and for � � 1; M� consists entirely of convex functions.

2.4 Certain subclasses of analytic functions of order

beta

In 1936, Robertson [92] introduced the concept of order of analytic functions. He de�ned

as follows.

De�nition 2.4.1 A function p is said to be in the class P (�) ; 0 � � < 1; if and only

if there exists a function p1 2 P such that

p (z) = (1� �) p1 (z) + �; z 2 E: (2.4.1)

That is, the class P (�) ; 0 � � < 1 consists of functions for which Re p (z) > � for

z 2 E and is the subclass of the class P:

Using the class P (�) ; 0 � � < 1; the classes of starlike and convex functions of order

� can be de�ned as follows.

S� (�) =

�f 2 A : zf

0 (z)

f (z)2 P (�) ; z 2 E

�;

C (�) =

�f 2 A : (zf

0 (z))0

f 0 (z)2 P (�) ; z 2 E

�:

For more details, see [20, 24, 92].

26

2.5 Subordination and di¤erential subordination

The concept of subordination was introduced by Lindelöf [41] and further enriched by

Littlewood [42, 43] and Rogosinski [93, 94]. The concept of subordination is heavily

depending on the Schwarz functions. So before going to the de�nition of subordination,

we give the de�nition of Schwarz function.

De�nition 2.5.1 Let � be the family of functions w; analytic in the open unit disk E

and satisfying the condition w (0) = 0; jw (z)j < 1 for z 2 E: Then the function w is

called the Schwarz function.

De�nition 2.5.2 If f and g are analytic in E, we say that f is subordinate to g; written

symbolically as f � g; if there exists a Schwarz function w in E such that

f (z) = g (w (z)) ; z 2 E:

If g is univalent in E, then the subordination is equivalent to f (0) = g (0) and f (E) �

g (E) : That is, f � g will mean that every value taken by f in E is also taken by g:

Derivatives play an important role in function theory, specially in geometrical analysis

of functions. Obtaining information about the properties of function from the properties

of its derivative is a signi�cant practice in functions of real variables. A large number

of di¤erential inequalities have been de�ned and studied by many renowned mathemati-

cians. Also in the theory of complex valued functions, there are several di¤erential

implications in which a characterization of a function is determined from a di¤erential

condition as we have seen in analytic conditions for starlikeness and convexity, de�ned

respectively by (2:3:1) and (2:3:2) :

De�nition 2.5.3 ([53]) A di¤erential subordination is merely a generalization of dif-

ferential inequality of real variable. For example

First order di¤erential subordination (p (z) ; zp0 (z) ; z) � h (z) ;

27

Second order di¤erential subordination �p (z) ; zp0 (z) ; z2p00 (z) ; z

�� h (z) :

De�nition 2.5.4 ([53]) Let : C3 � E �! C and let h be univalent in E. If p is

analytic in E and satis�es the di¤erential subordination (second order)

�p (z) ; zp0 (z) ; z2p00 (z) ; z

�� h (z) ; (2.5.1)

then p is called a solution of the di¤erential subordination. The univalent function q

is called a dominant of the solutions of the di¤erential subordination, or more simply a

dominant, if p � q for all p satisfying (2:5:1). A dominant ~q that satis�es ~q � q for all

dominants q of (2:5:1) is said to be the best dominant of (2:5:1).

2.6 Conic domains and associated functions

In this section, we shall discuss about the main component of this work, called conic

domains and circular domains will also be included in second part of this section. Some-

how circle is a special case of ellipse but here we shall see that circular domains are not

acting as the special case of general conic domains. That�s why, we shall discuss circular

domains in a separate way.

2.6.1 Hyperbolic, parabolic and elliptic domains

In 1991, Goodman [22, 23] initiated the concept of conic domain but unintentionally. It

is well known that for any convex function f (z) ; not only f (E) but also the images of

all circles lying in E centered at origin are convex arcs. Pinchuk raised a question that

whether this property is still valid for circles centered at other points. Goodman [22]

replied in negative answer and introduced the class of functions which obey this property.

He named such functions as uniformly convex functions and denoted the class of such

functions by UCV: He also introduced the class UST of uniformly starlike functions

which map the circles lying in E centered at other points onto star shaped arcs, see [23].

28

He not only just de�ned these classes but also gave the analytic conditions for uniform

convexity and uniform starlikeness. By using these analytic conditions, these classes can

be de�ned as follows.

De�nition 2.6.1 A function f 2 A is said to be in the class UCV of uniformly convex

functions, if

Re

�1 + (z � �)

f 00 (z)

f 0 (z)

�> 0; z; � 2 E:

De�nition 2.6.2 A function f 2 A is said to be in the class UST of uniformly starlike

functions, if

Re

�(z � �) f 0 (z)

f (z)� f (�)

�> 0; z; � 2 E:

We see that when we take � = 0; these classes coincide with the classes C and S�

respectively. From above de�nitions of UCV and UST , we see that famous Alexander�s

relation is failed to hold between UCV and UST: This fact can also be illustrated by

two counter examples, given in [22]. Rønning [97] introduced the class

ST = ff 2 A : f (z) = zg0 (z) ; g 2 UCV g

which is associated with the class UCV by Alexander type relation. He further proved

in [97] that neither UST ST nor ST UST: Later, again Rønning [95], and Ma

and Minda [51] independently gave the most suitable one variable characterization of the

class UCV and de�ned it as follows.

De�nition 2.6.3 A function f 2 A is said to be in the class UCV of uniformly convex

functions, if

Re

�1 +

zf 00 (z)

f 0 (z)

�>

����zf 00 (z)f 0 (z)

���� ; z 2 E:Similarly, the class ST was de�ned in the following form.

De�nition 2.6.4 A function f 2 A is said to be in the class ST of corresponding starlike

functions, if

Re

�zf 0 (z)

f (z)

�>

����zf 0 (z)f (z)� 1���� ; z 2 E:

29

This one variable characterization gave birth to the �rst conic (parabolic) domain

= fw : Rew > jw � 1jg :

In 1999, Kanas and Wisniowska [36, 35] generalized the above parabolic domain and

introduced the conic domain k; k � 0 and studied it comprehensively. This domain is

de�ned as

k =

�u+ iv : u > k

q(u� 1)2 + v2

�: (2.6.1)

This domain represents the right half plane for k = 0; hyperbolic regions (right branch)

when 0 < k < 1; a parabolic region for k = 1 and elliptic regions when k > 1 as shown

in Figure 2.2 below.

Figure 2.2: Boundaries of conic regions de�ned by k

30

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

given as:

pk(z) =

8>>>>>>>>><>>>>>>>>>:

1+z1�z , k = 0,

1 + 2�2

�log 1+

pz

1�pz

�2, k = 1,

1 + 21�k2 sinh

2��

2�arccos k

�arctanh

pz�, 0 < k < 1,

1 + 1k2�1 sin

0@ �2R(t)

u(z)ptR

0

1p1�x2

p1�(tx)2

dx

1A+ 1k2�1 , k > 1,

(2.6.2)

where u(z) = z�pt

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

��R0(t)4R(t)

�, R(t)

is the Legendre�s complete elliptic integral of the �rst kind and R0(t) is complementary

integral of R(t), for more detail, see [36, 35]. If pk(z) = 1 + �kz + � � � ; then it is shown

in [32] that from (2:6:2) ; one can have

�k =

8>>><>>>:8(arccos k)2

�2(1�k2) ; 0 � k < 1;

8�2; k = 1;

�2

4(k2�1)pt(1+t)R2(t)

; k > 1:

(2.6.3)

These conic regions are being studied by several authors, for example see [4, 5, 6, 59, 75,

83, 101]. Kanas and Wisniowska [36] de�ned the class P (pk) of functions which map the

open unit disk E onto these conic regions as follows.

De�nition 2.6.5 A function p (z) such that p (0) = 1; is said to be in the class P (pk)

if it subordinates to pk (z) with z 2 E: That is, p (E) � pk (E) = k:

Moreover, a function p (z) from class P (pk) possesses the following properties, see

[36].

1. Re p (z) > kk+1

: That is, P (pk) � P�

kk+1

�; the class of functions with real part

greater than kk+1

:

31

2. jarg p (z)j <

8<: arctan 1k; 0 < k <1;

�2; k = 0:

Kanas and Wisniowska [36, 35] also de�ned and studied the class k � UCV of k-

uniformly convex functions and the corresponding class k � ST of k-starlike functions.

These classes were de�ned subject to the conic domain k; k � 0.

The classes k � UCV and k � ST are de�ned as follows:

De�nition 2.6.6 A function f (z) 2 A is said to be in the class k � UCV , if and only

if,(zf 0(z))0

f 0 (z)2 P (pk) ;

that is,(zf 0(z))0

f 0 (z)� pk(z); z 2 E; k � 0;

or equivalently,

Re

�(zf 0(z))0

f 0 (z)

�> k

����(zf 0(z))0f 0 (z)� 1���� ; k � 0:

De�nition 2.6.7 A function f (z) 2 A is said to be in the class k � ST , if and only if,

zf 0(z)

f (z)2 P (pk) ;

that is,zf 0(z)

f (z)� pk(z); z 2 E; k � 0; (2.6.4)

or equivalently,

Re

�zf 0(z)

f (z)

�> k

����zf 0(z)f (z)� 1���� ; k � 0:

The famous Alexander relation holds between these classes, in fact, the class k � ST

was de�ned from k � UCV by mean of Alexander relation as

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

32

That�s why it is called the corresponding class of k�UCV: Now we include some results

related to these classes which we will use in subsequent chapters as special cases.

Theorem 2.6.1 [36] A function f 2 A and of the form (2:1:1) is in the class k�UCV;

if it satis�es the condition

1Xn=2

n fn+ k (n� 1)g janj < 1; k � 0: (2.6.5)

Theorem 2.6.2 [35] A function f 2 A and of the form (2:1:1) is in the class k�ST; if

it satis�es the condition

1Xn=2

fn+ k (n� 1)g janj < 1; k � 0: (2.6.6)

Theorem 2.6.3 [36] Let f (z) 2 S: Then f (z) 2 k � UCV for jzj < r0; where

r0 =1

2 (k + 1) +p4k2 + 6k + 3

=2 (k + 1)�

p4k2 + 6k + 3

2k + 1:

The conic domain k was then generalized to G (k; �) ; k � 0; 0 � � < 1 by Shams

et al. [101] and is de�ned as

G (k; �) = fw : Rew > k jw � 1j+ �g : (2.6.7)

Subject to this conic domain, the classesKD (k; �) and SD (k; �) are de�ned and studied

by Shams et al. [101]. These classes are de�ned as follows.

De�nition 2.6.8 A function f (z) from A is said to be a member of the class KD (k; �)

if

Re

�(zf 0(z))0

f 0 (z)

�> k

����(zf 0(z))0f 0 (z)� 1����+ �; k � 0; 0 � � < 1:

Similarly, by mean of Alexander relation, the class SD (k; �) is de�ned as

SD (k; �) = fg : g (z) = zf 0 (z) ; f (z) 2 KD (k; �)g :

33

Shams et al. [101] also gave the following su¢ cient conditions for these classes.

Theorem 2.6.4 [101] A function f 2 A and of the form (2:1:1) is in the classKD (k; �) ;

if it satis�es the condition

1Xn=2

n fn (k + 1)� (k + �)g janj < 1� �;

where k � 0; 0 � � < 1:

Theorem 2.6.5 [101] A function f 2 A and of the form (2:1:1) is in the class SD (k; �) ;

if it satis�es the condition

1Xn=2

fn (k + 1)� (k + �)g janj < 1� �;

where k � 0; 0 � � < 1:

The domain G (k; �) is a generalization of k with regard to order � (0 � � < 1).

Noor [59] generalized the conic domain k with regard to order � on the other way. She

de�ned the conic domain k;� as

k;� = (1� �) k + �; 0 � � < 1:

2.6.2 Circular domains

It was Janowski [30] who introduced the circular domain in 1973 by de�ning Janowski

functions. He de�ned these functions as:

De�nition 2.6.9 A function h (z) is said to be in the class P [A;B] if it is analytic in

E with h (0) = 1 and

h (z) � 1 + Az

1 +Bz; � 1 � B < A � 1:

34

Geometrically, a function h (z) 2 P [A;B] maps the open unit disk E onto the domain

[A;B] de�ned by

[A;B] =

�w :

����w � 1� AB

1�B2

���� < A�B

1�B2

�: (2.6.8)

This domain represents an open circular disk centered on the real axis with diameter end

points D1 =1�A1�B and D2 =

1+A1+B

with 0 < D1 < 1 < D2: The class P [A;B] is connected

with the class P of functions with positive real parts by the relation

h (z) 2 P , (A+ 1)h (z)� (A� 1)(B + 1)h (z)� (B � 1) 2 P [A;B] : (2.6.9)

It is known [66] that P [A;B] is a convex set and also noted that P [1;�1] = P; the class

of functions with positive real parts and P [1� 2�;�1] = P (�) ; 0 � � < 1; the class of

functions with real part greater than �:

Janowski [30] also de�ned the classes C [A;B] and S� [A;B] of Janowski convex and

Janowski starlike functions as follows.

De�nition 2.6.10 A function f 2 A is said to be the class C [A;B], if and only if

(zf 0 (z))0

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

De�nition 2.6.11 A function f 2 A is said to be the class S� [A;B], if and only if

zf 0 (z)

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

It is noted that the famous Alexander�s relation also holds for these classes, that is

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

These Janowski functions are being studied by several renowned mathematicians like

Noor [67, 68, 71, 72], Polato¼glu [88, 89], Cho [13, 14, 15, 16], Liu[44, 45, 46, 47, 48, 49]

35

etc.

The circular domain [A;B] was then generalized to [A;B; �] ; �1 � B < A �

1; 0 � � < 1 by Polato¼glu et al. [89]. This generalized domain is de�ned as

[A;B; �] =

�w :

����w � 1� [(1� �)A+ �B]B

1�B2

���� < (1� �) (A�B)

1�B2

�:

The class of functions which map the unit disk E onto the circular domain [A;B; �]

was named as P [A;B; �] : It is de�ned as follows.

De�nition 2.6.12 A function h (z) is said to be in the class P [A;B; �] if it is analytic

in E with h (0) = 1 and

h (z) � (1� �)1 + Az

1 +Bz+ �

=1 + [(1� �)A+ �B] z

1 +Bz; � 1 � B < A � 1; 0 � � < 1:

It is noted that P [A;B; 0] � P [A;B] and P [1;�1; 0] � P:

2.7 The class Vm of functions with bounded bound-

ary rotation and related classes

2.7.1 The class Vm of functions with bounded boundary rotation

It was Löwner [50] who initiated the concept of functions with bounded boundary rotation

but its extensive study was done by Paatero [84, 85]. Paatero explored the class Vm of

functions with bounded boundary rotations in following systematic way.

De�nition 2.7.1 A function f 2 A is said to be in the class Vm; if the variation of

36

tangent angle at the boundary of f (E) is at most m�; m � 2: That is, for m � 2;

2�Z0

����Re (zf 0 (z))0f 0 (z)

���� d� � m�; z 2 E;

Paatero [84, 85] also proved that for 2 � m � 4; the functions from Vm are univalent

also, that is, Vm � S; 2 � m � 4 and for m > 4; Vm consists of non-univalent functions

with radius of univalence ru = tan��m

�; proved by Kirwan [37]. It can easily be seen

that V2 consists of only convex univalent functions, that is, V2 � C: The class Vm has

been appearing as the ultimate part of advance research in this �eld of mathematics.

Noor wrote a number of article on its properties and applications, for example, see

[61, 64, 69, 76].

2.7.2 The class Rm of functions with bounded radius rotation

The class Rm of functions with bounded radius rotations was introduced by Tammi [106].

He de�ned it as follows.

De�nition 2.7.2 A function f 2 A is said to be in the class Rm; if for m � 2;

2�Z0

����Re zf 0 (z)f (z)

���� d� � m�; z 2 E;

It is noted that R2 consists of only starlike univalent functions, that is, R2 � S�: Also

the Alexander type relation holds between the classes Vm and Rm; that is

f 2 Vm () zf 0 2 Rm:

Noor wrote a number of article on its properties and applications, for example, see [78, 77].

37

2.7.3 The class Pm

In 1971, Pinchuk [87] introduced the class Pm of functions with bounded turnings. He

de�ned these functions as follows.

De�nition 2.7.3 A function p is said to be in the class Pm; if it is analytic in E satis-

fying p (0) = 1 and

p (z) =1

2�

2�Z0

1 + ze�it

1� ze�itd� (t) ; for every z 2 E;

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

that, for m � 2;2�Z0

d� (t) = 2 and

2�Z0

jd� (t)j � m

or equivalently, if p is analytic in E satisfying p (0) = 1 and

2�Z0

jRe p (z)j d� � m�; m � 2:

This class Pm is connected with the class P of functions with positive real part by

the following relation.

For p (z) 2 Pm; we have

p (z) =

�m

4+1

2

�p1 (z)�

�m

4� 12

�p2 (z) ; (2.7.1)

where p1 (z) ; p2 (z) 2 P: It can easily be seen that P2 � P:

Using this class, the above classes Vm and Rm can also be de�ned as:

Vm =

�f 2 A : (zf

0 (z))0

f 0 (z)2 Pm

�;

Rm =

�f 2 A : zf

0 (z)

f (z)2 Pm

�:

38

2.8 Ruscheweyh derivative

Before going to discuss about Ruscheweyh derivative, we need the concept of convolution

which is de�ned as follows.

De�nition 2.8.1 Let f (z) =P1

n=0 anzn and g (z) =

P1n=0 bnz

n be two power series,

convergent in E: Then convolution or Hadamard product of these series, denoted by f �g;

is de�ned as

(f � g) (z) =1Xn=0

anbnzn; z 2 E:

In 1975, using the concept of convolution, Ruscheweyh [99] introduced a linear oper-

ator D� : A �! A: it is de�ned as

D�f (z) =z

(1� z)�+1� f (z)

= z +1Xn=2

'n (�) an zn; (� > �1) (2.8.1)

with

'n (�) =(� + 1)n�1(n� 1)! ; (2.8.2)

where (:)n is a Pochhammer symbol given as

(�)n =

8<: 1; n = 0;

� (�+ 1) (�+ 2) : : : (�+ n� 1) ; n 2 N:(2.8.3)

Moreover; we note that D0f (z) = f (z) ; D1f (z) = zf 0 (z) and

Dnf (z) =z (zn�1f (z))

(n)

n!; n 2 N0 = f0; 1; 2; :::g :

The symbol Dnf is called nth order Ruscheweyh derivative of the function f: The fol-

lowing identity can easily be established.

39

For a real number � (� > �1) ; we have

z�D�f (z)

�0= (� + 1)D�+1f (z)� �D�f (z) : (2.8.4)

For the applications of Ruscheweyh derivative; see [58, 60, 62, 63].

2.9 Hypergeometric functions

Hypergeometric functions are special functions, obtained as the solution of special types

of di¤erential equations. We discuss the following two types of hypergeometric functions.

2.9.1 Con�uent hypergeometric function

Let a and c be complex numbers with c 6= 0;�1;�2; :::. Then the con�uent (or Kummer)

hypergeometric function, denoted by 1F1 (a; c; z) or � (a; c; z) ; is de�ned as

1F1 (a; c; z) = 1 +a

c

z

1!+a (a+ 1)

c (c+ 1)

z2

2!+ :::

=1Xk=0

(a)k(c)k

zk

k!;

where (:)k is the Pochhammer symbol given by (2:8:3) :The following are some properties

of con�uent hypergeometric functions which will be used in our subsequent work.

Remark 2.9.1 [53] For real or complex numbers a; c (c 6= 0;�1;�2; : : :) and Re c >

Re a > 0; we have

1Z0

ta�1 (1� t)c�a�1 etzdt =� (a) � (c� a)

� (c)1F1 (a; c; z) ; (2.9.1)

1F1 (a; c; z) = ez 1F1 (c� a; c;�z) : (2.9.2)

40

2.9.2 Gaussian hypergeometric function

Let a; b and c be complex numbers with c 6= 0;�1;�2; :::. Then the Gaussian hyperge-

ometric function, denoted by 2F1 (a; b; c; z) or F (a; b; c; z) ; is de�ned as

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

c

z

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

c (c+ 1)

z2

2!+ :::

=

1Xk=0

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

zk

k!;

where (:)k is the Pochhammer symbol given by (2:8:3) : The following are some properties

of Gaussian hypergeometric functions which will be used in our subsequent work.

Remark 2.9.2 [53] For real or complex numbers a; b; c (c 6= 0;�1;�2; : : :) and Re c >

Re b > 0; we have

1Z0

tb�1 (1� t)c�b�1 (1� tz)�a dt =� (b) � (c� b)

� (c)2F1 (a; b; c; z) ; (2.9.3)

2F1 (a; b; c; z) = (1� z)�a 2F1

�a; c� b; c;

z

z � 1

�; (2.9.4)

2F1 (a; b; c; z) = 2F1 (b; a; c; z) : (2.9.5)

For more details of hypergeometric functions, we refer to [53].

2.10 Preliminary results

We need the following lemmas in our main results of subsequent chapters.

Lemma 2.10.1 [33] Let 0 � k < 1: Also, let �; � 2 C be such that � 6= 0 and

Re (�k= (k + 1) + �) > 0: If p (z) is analytic in E; p (0) = 1; p (z) satis�es

p (z) +zp0 (z)

�p (z) + �� pk (z) ; (2.10.1)

41

and q (z) is analytic solution of

q (z) +zq0 (z)

�q (z) + �= pk (z) ;

then q (z) is univalent, p (z) � q (z) � pk (z) ; and q (z) is best dominant of (2:10:1) :

Moreover, the solution q (z) is given by

q (z) =

��

Z 1

0

�t�+��1 exp

Z tz

z

pk (u)� 1u

du

��dt

��1� �

�:

For more details of best dominant, we refer to [33, 53].

Lemma 2.10.2 If f (z) � H (z) and g (z) � H (z) ; then for t 2 [0; 1] ;

(1� t) f (z) + tg (z) � H (z) :

Lemma 2.10.3 [29] Let the function w (z) be non-constant analytic in E with w (0) = 0:

If jw (z)j attains its maximum value on the circle jzj = r < 1 at a point z0; then

z0w0 (z0) = cw (z0) ;

c is real and c � 1:

Lemma 2.10.4 [53] Let u = u1 + iu2, v = v1 + iv2 and (u; v) be a complex valued

function satisfying the conditions:

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

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

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

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

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

42

Lemma 2.10.5 [94] Let h (z) = 1 +1Pn=1

cnzn be subordinate to H (z) = 1 +

1Pn=1

Cnzn: If

H (z) is univalent in E and H (E) is convex, then

jcnj � jC1j ; n � 1:

43

Chapter 3

On Generalized �-Convex Functions Associated with

Conic Domain

44

The classM� of �- convex functions is de�ned in De�nition 2.3.3 and a detailed overview

of conic domains along with the relevant classes of functions is given in section 2.6.1. It

is evident from the de�nition of �-convex functions, convex and starlike functions can

be obtained as special cases of �-convex functions. The class UM (�; k) of �-convex

functions associated with conic domains is introduced by Kanas [31] and this class gives

k-uniformly convex and corresponding k-starlike functions as special cases. This class is

de�ned by subordinating the function

J (�; f (z)) = (1� �)zf 0 (z)

f (z)+ �

(zf 0 (z))0

f 0 (z)

to the extremal function pk (z) given by (2:6:2) : Also the classM� of �- convex functions

is obtained as the special case of UM (�; k) as UM (�; 0) �M�:

In this chapter, we will generalize the class UM (�; k) by generalizing the function

J (�; f (z)) and introduce the generalized class k � UB (�; �; ) : This class gives several

known classes as special cases. Many interesting properties of this class will be investi-

gated here and many known results can be deduced from our main results, we will show

this fact as corollaries.

It is also to be mentioned here that all the contents of this chapter have been published

in "Mathematical and Computer Modelling, Vol 55; 2012; Pages 844� 852", see [74].

3.1 Introduction

Let S� (�) ; C (�) denote the well-known classes of starlike and convex functions of order

� (0 � � < 1) respectively as discussed in section 2.4. The class k�UCV of k-uniformly

convex functions and corresponding class k � ST of k-starlike functions are introduced

and studied by Kanas and Wisniowska [36, 35] and discussed in section 2.6.1. These

classes were generalized to KD (k; �) and SD (k; �) respectively by Shams et al. [101]

as discussed in section 2.6.1.

Now we de�ne the following.

45

De�nition 3.1.1 A function f (z) 2 A is said to be in the class k�UB (�; �; ) ; k � 0;

if and only if, for � � 0; 0 � � < 1; 0 � < 1;

Re J (�; �; ; f (z)) > k jJ (�; �; ; f (z))� 1j ; (3.1.1)

where

J (�; �; ; f (z)) =1� �

1� �

�zf 0 (z)

f (z)� �

�+

�

1�

�1� +

zf 00 (z)

f 0 (z)

�: (3.1.2)

Special Cases

(i) k�UB (1; �; 0) = k�UCV and k�UB (0; 0; ) = k� ST; the well-known classes

of k-uniformly convex and k-starlike functions respectively, introduced by Kanas

and Wisniowska [36, 35].

(ii) k � UB (0; �; ) = SD (k; �) and k � UB (1; �; ) = KD (k; ) ; the well-known

classes, introduced and studied in [101].

(iii) k�UB (�; 0; 0) = UM (�; k) ; the well-known class, introduced and studied in [31].

(iv) 0� UB (�; 0; 0) =M�; the well-known class of alpha-convex functions, introduced

and studied in [54].

Similarly, several more special cases can be obtained from k � UB (�; �; ) by giv-

ing particular values to di¤erent parameters. That are 1 � UB (0; 0; ) = ST; 1 �

UB (1; �; 0) = UCV; 0�UB (0; 0; ) = S�; 0�UB (1; �; 0) = C; 0�UB (0; �; ) = S� (�)

and 0� UB (1; �; ) = C ( ) : For detail of these classes, see [22, 24, 97].

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

only if, the function J (�; �; ; f (z)) takes all values in the conic domain k; given by

(2:6:1) : Taking this geometrical interpretation into consideration, one can rephrase the

above de�nition as:

46

A function f (z) 2 A is said to be in the class k�UB (�; �; ) ; k � 0; if and only if,

for � � 0; 0 � � < 1; 0 � < 1;

J (�; �; ; f (z)) � pk (z) ; (3.1.3)

where pk (z) is de�ned by (2:6:2) :

3.2 Main results

Our �rst result is a su¢ cient condition for a function f (z) 2 A to be from k�UB (�; �; ) :

It gives many known results as special cases as discussed after this result.

Theorem 3.2.1 A function f (z) of the form (2:1:1) is in the class k � UB (�; �; ) if

1Xn=2

n (k;�; �; ) < (1� �) (1� ) ; (3.2.1)

where

n (k;�; �; )

= (k + 1) f(n� 1) (1� �) (1� ) + n� (1� �) (n� 1)g janj

+ (k + 1)n�1Xj=2

f(j � 1) (1� �) (1� ) + � (1� �) (n� j)g (n+ 1� j) jajan+1�jj

+ (1� �) (1� ) (n+ 1) janj+ (1� �) (1� )n�1Xj=2

(n+ 1� j) jajan+1�jj :

Proof. Assuming that (3:2:1) holds, then it su¢ ces to show that

k jJ (�; �; ; f (z))� 1j � Re fJ (�; �; ; f (z))� 1g < 1:

47

Now consider

jJ (�; �; ; f (z))� 1j

=

����1� �

1� �

�zf 0 (z)

f (z)� �

�+

�

1�

�1� +

zf 00 (z)

f 0 (z)

�� 1����

=

����1� �

1� �

zf 0 (z)

f (z)� (1� �) �

1� �+ �+

�

1�

zf 00 (z)

f 0 (z)� 1����

=

����1� �

1� �

zf 0 (z)

f (z)� 1� �

1� �+

�

1�

zf 00 (z)

f 0 (z)

����

=

������������

(1� �) (1� ) zf 0 (z) f 0 (z)

� (1� �) (1� ) f (z) f 0 (z) + � (1� �) zf (z) f 00 (z)

(1� �) (1� ) f (z) f 0 (z)

������������: (3.2.2)

Now from (2:1:1) ; we have

zf 0 (z) f 0 (z) = z

1Xn=0

nanzn�1

! 1Xn=0

nanzn�1

!; a0 = 0; a1 = 1

=1

z

1Xn=0

nanzn

! 1Xn=0

nanzn

!

=1

z

1Xn=0

nXj=0

j (n� j) ajan�j

!zn

=1Xn=0

nXj=0

j (n� j) ajan�j

!zn�1

= z +

1Xn=3

nXj=0

j (n� j) ajan�j

!zn�1

48

= z +1Xn=2

n+1Xj=0

j (n+ 1� j) ajan+1�j

!zn

= z +1Xn=2

2nan +

n�1Xj=2

j (n+ 1� j) ajan+1�j

!zn:

Similarly, we can have

f (z) f 0 (z) = z +

1Xn=2

(n+ 1) an +

n�1Xj=2

(n+ 1� j) ajan+1�j

!zn

and

zf (z) f 00 (z) =1Xn=2

n (n� 1) an +

n�1Xj=2

(n+ 1� j) (n� j) ajan+1�j

!zn:

Using the above equalities in (3:2:2) ; we have

jJ (�; �; ; f (z))� 1j

=

����������������������

1Pn=2

[f2n (1� �) (1� )� (n+ 1) (1� �) (1� )

+n� (1� �) (n� 1)g an +n�1Pj=2

fj (1� �) (1� )

� (1� �) (1� ) + � (1� �) (n� j)g (n+ 1� j) ajan+1�j] zn

(1� �) (1� )

(z +

1Pn=2

(n+ 1) an +

n�1Pj=2

(n+ 1� j) ajan+1�j

!zn

)

����������������������

49

�

1Pn=2

j(n� 1) (1� �) (1� ) + n� (1� �) (n� 1)j janj

+1Pn=2

�����n�1Pj=2 f(j � 1) (1� �) (1� ) + � (1� �) (n� j)g (n+ 1� j) ajan+1�j

�����(1� �) (1� )

(1�

1Pn=2

(n+ 1) janj �1Pn=2

�����n�1Pj=2 (n+ 1� j) ajan+1�j

�����) :

(3.2.3)

Now from (3:2:3) ; we have

k jJ (�; �; ; f (z))� 1j � Re fJ (�; �; ; f (z))� 1g

� (k + 1) jJ (�; �; ; f (z))� 1j

�

(k + 1)1Pn=2

j(n� 1) (1� �) (1� ) + n� (1� �) (n� 1)j janj

+(k + 1)1Pn=2

�����n�1Pj=2 f(j � 1) (1� �) (1� ) + � (1� �) (n� j)g (n+ 1� j) ajan+1�j

�����(1� �) (1� )

(1�

1Pn=2

(n+ 1) janj �1Pn=2

�����n�1Pj=2 (n+ 1� j) ajan+1�j

�����) :

The last expression is bounded by 1 if

1Xn=2

[(k + 1) j(n� 1) (1� �) (1� ) + n� (1� �) (n� 1)j janj

+ (k + 1)n�1Xj=2

jf(j � 1) (1� �) (1� ) + � (1� �) (n� j)g (n+ 1� j) ajan+1�jj

+(1� �) (1� ) (n+ 1) janj+ (1� �) (1� )

n�1Xj=2

(n+ 1� j) jajan+1�jj#

< (1� �) (1� ) :

This completes the proof.

50

Corollary 3.2.1 When � = 0; then (3:2:1) reduces to

(1� �) (1� )

>1Xn=2

((k + 1) (n� 1) (1� ) janj+ (k + 1)

n�1Xj=2

(j � 1) (1� ) (n+ 1� j) jajan+1�jj

+(1� �) (1� ) janj+ (1� �) (1� )n�1Xj=1

(n+ 1� j) jajan+1�jj)

> (1� )1Xn=2

f(k + 1) (n� 1) + (1� �)g janj :

This implies that1Xn=2

fn (k + 1)� (k + �)g janj < 1� �;

which is the su¢ cient condition for f (z) 2 SD (k; �) ; proved in [101].

Corollary 3.2.2 When � = 1; then (3:2:1) reduces to

(1� �) (1� )

>1Xn=2

(n (k + 1) (n� 1) (1� �) janj+ (k + 1)

n�1Xj=2

(1� �) (n� j) (n+ 1� j) jajan+1�jj

+n (1� �) (1� ) janj+ (1� �) (1� )

nXj=2

(n+ 1� j) jajan+1�jj)

> (1� �)

1Xn=2

n f(k + 1) (n� 1) + (1� )g janj :

51

This implies that1Xn=2

n fn (k + 1)� (k + )g janj < 1� ;

which is the su¢ cient condition for f (z) 2 KD (k; ) ; proved in [101].

When � = 0; = 0; then Theorem 3.2.1 gives the following su¢ cient condition for

f (z) 2 UM (�; k) ; the class introduced by Kanas [31].

Corollary 3.2.3 A function f (z) of the form (2:1:1) is in the class UM (�; k) if

1Xn=2

n (k;�) < 1; (3.2.4)

where

n (k;�) = (k + 1) (n� 1) (1� �+ n�) janj

+(k + 1)n�1Xj=2

f(j � 1) (1� �) + � (n� j)g (n+ 1� j) jajan+1�jj

+(n+ 1) janj+n�1Xj=2

(n+ 1� j) jajan+1�jj :

Corollary 3.2.4 When � = 1; = 0; then (3:2:1) reduces to

1Xn=2

n fn+ k (n� 1)g janj < 1;

which is the su¢ cient condition for f (z) 2 k � UCV; proved in [36].

Corollary 3.2.5 When � = 0; � = 0; then (3:2:1) reduces to

1Xn=2

fn+ k (n� 1)g janj < 1;

which is the su¢ cient condition for f (z) 2 k � ST; proved in [35].

52

Corollary 3.2.6 When � = 0; k = 0; then (3:2:1) reduces to

1Xn=2

(n� �) janj < 1� �;

which is the su¢ cient condition for f (z) 2 S� (�) ; proved in [100].

Similarly, several more results can also be deduced from Theorem 3.2.1 by giving

particular values to the parameters.

The following is an inclusion result stating the fact that k�UB (�; �; ) � SD (k; �)

for some � = � (�; �; ) de�ned below.

Theorem 3.2.2 Let f (z) 2 k � UB (�; �; ) : Then f (z) 2 SD (k; �) ; where

� = 1� (1� �) (1� )

(1� �) (1� ) + � (1� �): (3.2.5)

Proof. Let1

1� �

�zf 0 (z)

f (z)� �

�= p (z) ; (3.2.6)

where p (z) is analytic in E and p (0) = 1: This implies that

zf 0 (z)

f (z)= (1� �) p (z) + �: (3.2.7)

Now di¤erentiating logarithmically, we have

1 +zf 00 (z)

f 0 (z)= (1� �) p (z) + � +

(1� �) zp0 (z)

(1� �) p (z) + �: (3.2.8)

53

Using (3:2:7) and (3:2:8) ; we have

J (�; �; ; f (z)) =1� �

1� �f(1� �) p (z) + � � �g

+�

1�

�(1� �) p (z) + � +

(1� �) zp0 (z)

(1� �) p (z) + ��

�

=

�1� �

1� �+

�

1�

�f(1� �) p (z) + �g

�(1� �) �

1� �� �

1� +

�

1�

(1� �) zp0 (z)

(1� �) p (z) + �:

Now for � = 1� (1��)(1� )(1��)(1� )+�(1��) ; we have

J (�; �; ; f (z)) = p (z) +1� �

1� �+

�

1� � 1

�(1� �) �

1� �� �

1� +

zp0 (z)1� �p (z) + �

1��

= p (z) +zp0 (z)

1� �p (z) + �

1��:

Since f (z) 2 k � UB (�; �; ) ; so we obtain

p (z) +zp0 (z)

1� �p (z) + �

1��� pk (z) ;

where pk (z) is de�ned by (2:6:2) : Since Ren1� �

kk+1

+ �1��

o> 0; z 2 E; therefore apply-

ing Lemma 2.10.1, we have

1

1� �

�zf 0 (z)

f (z)� �

�= p (z) � pk (z) ;

which implies that f (z) 2 SD (k; �) :

By giving particular values to di¤erent parameters in Theorem 3.2.2 we get the fol-

lowing well-known result, proved by Mocanu in [54].

54

Corollary 3.2.7 Let f (z) 2 0� UB (�; 0; 0) = M�: Then f (z) 2 SD (0; 0) = S�: That

is,

M� � S�; � � 0:

The following is the integral representation of functions from k � UB (�; �; ) :

Theorem 3.2.3 A function f (z) 2 k � UB (�; �; ) ; � 6= 0; if and only if, there is a

function g (z) 2 k � ST such that

f (z) =

24� zZ0

t��1�g (t)

t

� 1� �

dt

35 1�

; (3.2.9)

where � = 1 + (1��)(1� )�(1��) :

Proof. From (3:2:9) ; we can have

1

�(f (z))� =

zZ0

t��1�g (t)

t

� 1� �

dt

which reduces to

(f (z))��1 f 0 (z) = z��1�g (z)

z

� 1� �

:

Logarithmic di¤erentiation leads us to

(� � 1) zf0 (z)

f (z)+zf 00 (z)

f 0 (z)= (� � 1) + 1�

�

�zg0 (z)

g (z)� 1�

which implies that

zg0 (z)

g (z)=� (� � 1)1�

zf 0 (z)

f (z)+

�

1�

zf 00 (z)

f 0 (z)� � (� � 1)

1� + 1:

For � = 1 + (1��)(1� )�(1��) ; we have

zg0 (z)

g (z)=1� �

1� �

zf 0 (z)

f (z)+

�

1�

zf 00 (z)

f 0 (z)+�� �

1� �

55

which implies that

zg0 (z)

g (z)=

1� �

1� �

�zf 0 (z)

f (z)� �

�+

�

1�

�1� +

zf 00 (z)

f 0 (z)

�= J (�; �; ; f (z)) :

Now from (3:1:3) and using (2:6:4), we obtain the required result.

Theorem 3.2.4 Let f (z) 2 k � UB (�; �; ) : Then the function

g (z) = z

�f (z)

z

� 1��1��

(f 0 (z))�1� (3.2.10)

belongs to k � ST for all z 2 E:

Proof follows immediately from logarithmic di¤erentiation of (3:2:10).

Theorem 3.2.5 For � > �1 � 0;

k � UB (�; �; ) � k � UB (�1; �; ) :

Proof. Let f (z) 2 k � UB (�; �; ) : Then consider

J (�1; �; ; f (z)) =1� �11� �

�zf 0 (z)

f (z)� �

�+

�11�

�1� +

zf 00 (z)

f 0 (z)

�

=�1� �1

�

� 1

1� �

�zf 0 (z)

f (z)� �

�+�1�

�1� �

1� �

�zf 0 (z)

f (z)� �

�+

�

1�

�1� +

zf 00 (z)

f 0 (z)

��

=�1� �1

�

�J (0; �; ; f (z)) +

�1�J (�; �; ; f (z)) :

Now as f (z) 2 k � UB (�; �; ) ; so

J (�; �; ; f (z)) � pk (z) ;

56

which implies by Theorem 3.2.2,

J (0; �; ; f (z)) � pk (z) :

Using these along with the Lemma 2.10.2, we have

J (�1; �; ; f (z)) � pk (z) ;

which implies that f (z) 2 k � UB (�1; �; ) :

For the function f (z) 2 A; we consider the integral operator

Ia (f) =a+ 1

za

zZ0

ta�1f (t) dt; a = 1; 2; 3; � � � : (3.2.11)

This operator Ia was introduced by Bernardi [9]. In particular, the operator I1 was

studied earlier by Libera [40]. Now we prove the following.

Theorem 3.2.6 Let f (z) 2 k�UB (�; �; ) : Then Ia (f) 2 SD (k; �) ; where � is de�ned

by (3:2:5) :

Proof. From (3:2:11) ; we have

(a+ 1) za�1f (z) = (za (Iaf (z)))0

= za (Iaf (z))0 + aza�1 (Iaf (z))

which implies that

(a+ 1) f (z) = z (Iaf (z))0 + aIaf (z) :

57

Di¤erentiating logarithmically, we have

zf 0 (z)

f (z)=

z�z (Iaf (z))

0�0 + za (Iaf (z))0

z (Iaf (z))0 + aIaf (z)

=

z(z(Iaf(z))0)0

Iaf(z)+ a z(Iaf(z))

0

Iaf(z)

z(Iaf(z))0

Iaf(z)+ a

: (3.2.12)

Let z(Iaf(z))0

Iaf(z)= (1� �) p (z) + � with p (z) analytic in E and p (0) = 1: Then

z (Iaf (z))0 = f(1� �) p (z) + �g Iaf (z) :

Di¤erentiating, we have

�z (Iaf (z))

0�0 = (1� �) p0 (z) (Iaf (z)) + f(1� �) p (z) + �g (Iaf (z))0

which implies that

z�z (Iaf (z))

0�0Iaf (z)

= (1� �) zp0 (z) + [(1� �) p (z) + �]2 :

Using this in (3:2:12) ; we have

zf 0 (z)

f (z)=

(1� �) zp0 (z) + [(1� �) p (z) + �]2 + a [(1� �) p (z) + �]

[(1� �) p (z) + �] + a

= (1� �) p (z) + � +(1� �) zp0 (z)

(1� �) p (z) + � + a

= (1� �)h (z) + �; (3.2.13)

where

h (z) = p (z) +zp0 (z)

(1� �) p (z) + � + a: (3.2.14)

58

Now from (3:2:13) ; we have

1 +zf 00 (z)

f 0 (z)= (1� �)h (z) + � +

(1� �) zh0 (z)

(1� �)h (z) + �: (3.2.15)

Using (3:2:13) and (3:2:15) ; we have

J (�; �; ; f (z)) =1� �

1� �f(1� �)h (z) + � � �g

+�

1�

�(1� �)h (z) + � +

(1� �) zh0 (z)

(1� �)h (z) + ��

�

=

�1� �

1� �+

�

1�

�f(1� �)h (z) + �g

�(1� �) �

1� �� �

(1� )+

�

1�

(1� �) zh0 (z)

(1� �)h (z) + �:

Now for � = 1� (1��)(1� )(1��)(1� )+�(1��) ; we have

J (�; �; ; f (z)) = h (z) +1� �

1� �+

�

1� � 1

�(1� �) �

1� �� �

(1� )+

zh0 (z)1� �h (z) + �

1��

= h (z) +zh0 (z)

1� �h (z) + �

1��:

Since f (z) 2 k � UB (�; �; ) ; so

h (z) +zh0 (z)

1� �h (z) + �

1��� pk (z) ;

where pk (z) is de�ned by (2:6:2) : Since Ren1� �

kk+1

+ �1��

o> 0; z 2 E; therefore apply-

ing Lemma 2.10.1, we have

p (z) +zp0 (z)

(1� �) p (z) + � + a= h (z) � pk (z) :

59

Now as Re�(1� �) k

k+1+ � + a

> 0; z 2 E; so again using Lemma 2.10.1, we have

p (z) � pk (z) ;

which implies that Iaf (z) 2 SD (k; �) :

From the above theorem, one can easily deduce the following known [101, 9] results.

Corollary 3.2.8 Let f (z) 2 k � UB (0; �; ) = SD (k; �) : Then Ia (f) 2 SD (k; �) :

Corollary 3.2.9 Let f (z) 2 k � UB (0; 0; ) = k � ST: Then Ia (f) 2 k � ST:

Corollary 3.2.10 Let f (z) 2 0� UB (0; 0; ) = S�: Then Ia (f) 2 S�:

3.3 Conclusion

We generalized the class UM (�; k) by generalizing the function J (�; f (z)) and intro-

duced the class k�UB (�; �; ) : Several known classes are obtained from k�UB (�; �; )

as special cases. Many interesting properties of this class are investigated and many

known results are deduced from our main results which are included as corollaries.

60

Chapter 4

On a New Class of Analytic Functions Associated

with Conic Domain

61

In the geometry of image domain of analytic functions, conic domains have got great

importance due to their convexity and symmetry about the positive real axis. The study

of conic domains was initiated by Goodman [22, 23] for a particular case (parabolic

domain) and later, general conic domain k; k � 0 (all three cases) which we have

discussed in section 2.6.1 in some detail, was comprehensively studied by Kanas and

Wisniowska [36, 35] but these conic regions were restricted to have �xed vertices and

lengths of latus rectum. These conic domains could neither be contracted nor magni�ed.

We have removed this de�ciency of conic domain k; k � 0 by introducing a gen-

eralized conic domain k (a; b) ; k � 0: This generalized conic domain gives contraction

as well as magni�cation of the conic domain k; k � 0: A number of conic regions of

any size can be obtained from our de�ned generalized conic domain k (a; b) ; k � 0 by

assigning suitable values to parameters a and b:

In this chapter, we will give a detailed overview of generalized conic domaink (a; b) ; k �

0 and discuss all of its aspects. Di¤erent graphical views of this generalized conic domain

for speci�c values of parameters will be shown for better understanding of the behaviour

of this domain. The class k � P (a; b) of functions which map the open unit disk E onto

these generalized conic regions will be de�ned and some of its properties will be discussed.

Related to the class k � P (a; b) ; two more classes k � UCV (a; b) and k � ST (a; b) will

be introduced and some results concerning to these classes will be investigated.

All contents of this chapter have been published in the well reputed journal "Com-

puters and Mathematics with Applications, Vol 62; 2011; Pages 367 � 375", for detail,

see [70].

4.1 Introduction

As we have mentioned above, we are to generalize the conic domain k; k � 0, introduced

by Kanas and Wisniowska [36, 35], discussed in section 2.6.1. we de�ne the following.

62

De�nition 4.1.1 A function p (z) is said to be in the class k � P (a; b) ; if and only if,

p (z) � pk (a; b; z) ; (4.1.1)

where k 2 [0;1) ;

pk (a; b; z) = 1 + a+ (1� b) fpk(z)� 1g (4.1.2)

= a+ b+ (1� b) pk(z);

and pk(z) is de�ned by (2:6:2) : Also a and b must be chosen accordingly, as:

(i) For k = 0; we take b = 0;

(ii) For k 2�0; 1p

2

�; we take b 2

�1

2k2�1 ; 1�;

(iii) For k 2h1p2; 1i; we take b 2 (�1; 1) ;

(iv) For k 2 (1;1) ; we take b 2��1; 1

2k2�1�:

9>>>>>>=>>>>>>;(4.1.3)

and

k2(1�b)1�k2 � � � a < 1� k2(1�b)

k2�1 + �; 0 � k < 1;

�1+b2� a < 1�b

2; k = 1;

max�k2(1�b)1�k2 � �; 1� k2(1�b)

k2�1 � ��� a < 1� k2(1�b)

k2�1 + �; k > 1;

9>>>>>>>>>=>>>>>>>>>;(4.1.4)

where � = kpk2(1�b)2+(1�k2)(1�b2)

k2�1 :

Geometrically, the function p (z) 2 k�P (a; b) takes all values from the conic domain

k (a; b) which is de�ned as:

k (a; b) =�u+ iv : (u� a)2 > k2

�(u� a+ b� 1)2 + v2 + 2b (1� b)

�: (4.1.5)

63

The conic domain k (a; b) represents the right half plane when k = 0; a hyperbola

(right branch) when 0 < k < 1; a parabola when k = 1 and an ellipse when k > 1: It

can be seen that k (0; 0) = k; the conic domain de�ned by Kanas and Wisniowska

[36, 35], consequently, k � P (0; 0) = P (pk) ; the well-known class introduced by Kanas

and Wisniowska [36]. The function p1 (a; b; z) = Qa;b (z) is de�ned by Kanas in [33]. Here

are some basic facts about the class k � P (a; b) :

Remark 4.1.1 1. k � P (a; b) � P (�) ; where

� =

8>>><>>>:a+ 1+b

2; k = 1;

a+k2(1�b)�k

pk2(1�b)2+(1�k2)(1�b2)k2�1 ; k 6= 1:

(4.1.6)

That is, for p (z) 2 k � P (a; b) we have Re p (z) > �; z 2 E where � is de�ned by

(4:1:6) :

2. k � P (a1; b) � k � P (a2; b) ; a1 > a2; k 2 [0; 1] :

3. k � P (a; b1) � k � P (a; b2) ; b1 > b2; k 2 (0;1) :

The domain k (a; b) always ensures that the point (1; 0) is contained inside it whereas

the domain k;�; introduced in [59], de�ned by

k;� = (1� �) k + �; 0 � � < 1; k � 0; (4.1.7)

is not well de�ned because (1; 0) =2 k;� in general. For example, in particular (1; 0) =2

1:2;0:5 as shown in �gure below.

64

Figure 4.1: View of 1:2;0:5

From (4:1:7) ; we note that (u; v) 2 k;� if

u2 > k2�(u+ � � 1)2 + v2 + 2� (1� �)

�(4.1.8)

and for (u; v) = (1; 0) ; k = 1:2; � = 0:5; the relation (4:1:8) reduces to 1 > 1:08 which

clearly shows that (1; 0) =2 1:2;0:5: We see that the conic domain k (0; b) coincides with

k;b only when b is chosen according to (4:1:3) : This means that for k;� to contain the

point (1; 0) inside, � must be chosen accordingly, as:

� 2

8>>><>>>:[0; 1) ; if 0 � k � 1;

h0; 1�

pk2�1k

�; if k > 1:

(4.1.9)

According to this criteria de�ned in (4:1:9) ; for k = 1:2; there must be � 2 [0; 0:4472292)

to contain the point (1; 0) inside.

The domain k;� gives only the contraction of k whereas the domain k (a; b) gives

contraction as well as magni�cation of k depending upon b: For b > 0; the domain

k (a; b) gives the contraction as shown in �gure below.

65

Figure 4.2: Contraction of k

For b < 0; the domain gives the magni�cation of k as can be seen from the �gure below.

Figure 4.3: Magni�cation of k

Now we de�ne the classes k � UCV (a; b) and k � ST (a; b) as follows.

De�nition 4.1.2 A function f (z) 2 A is said to be in the class k�UCV (a; b) ; k � 0;

a; b satisfy (4:1:3) and (4:1:4) ; if and only if ,

�Re

�(zf 0(z))0

f 0 (z)� a

��2> k2

"����(zf 0(z))0f 0 (z)� a+ b� 1

����2 + 2b (1� b)

#(4.1.10)

66

or equivalently(zf 0(z))0

f 0 (z)� pk (a; b; z) ; (4.1.11)

where pk (a; b; z) is de�ned by (4:1:2) :

De�nition 4.1.3 A function f (z) 2 A is said to be in the class k � ST (a; b) ; k � 0;

a; b satisfy (4:1:3) and (4:1:4) ; if and only if ,

�Re

�zf 0(z)

f (z)� a

��2> k2

"����zf 0(z)f (z)� a+ b� 1

����2 + 2b (1� b)

#(4.1.12)

or equivalentlyzf 0(z)

f (z)� pk (a; b; z) ; (4.1.13)

where pk (a; b; z) is de�ned by (4:1:2) :

It can be easily seen that

f (z) 2 k � UCV (a; b) () zf 0 (z) 2 k � ST (a; b) :

Special Cases

(i) k � UCV (0; 0) = k � UCV; the well-known class of k-uniformly convex functions,

introduced by Kanas and Wisniowska [36].

(ii) k � ST (0; 0) = k � ST; the well-known class of k-starlike functions, introduced by

Kanas and Wisniowska [35].

4.2 Main results

The following �rst result is the su¢ cient condition for a function f (z) 2 A to be from

the class k � ST (a; b) :

67

Theorem 4.2.1 If f (z) 2 A satis�es the inequality

Re

( zf 00(z)f 0(z)

zf 0(z)f(z)

� 1

)<3� �

2� �;

where � is de�ned by (4:1:6) ; then f (z) 2 k � ST (a; b) ; k 2 [0; 1] ; b � 0 with a and b

satisfying (4:1:3) and (4:1:4) :

Proof. We consider the function w (z) as

zf 0 (z)

f (z)� 1 = (1� �)w (z) ; (4.2.1)

where � is de�ned by (4:1:6) : We see that w (z) is analytic in E and w (0) = 0: Loga-

rithmic di¤erentiation of (4:2:1) gives us

1 +zf 00 (z)

f 0 (z)� zf 0 (z)

f (z)=

(1� �) zw0 (z)

(1� �)w (z) + 1:

This implies thatzf 00 (z)

f 0 (z)= (1� �)w (z) +

(1� �) zw0 (z)

(1� �)w (z) + 1: (4.2.2)

Now from (4:2:1) and (4:2:2) ; we have

zf 00(z)f 0(z)

zf 0(z)f(z)

� 1= 1 +

zw0 (z)

w (z) f(1� �)w (z) + 1g :

Suppose that there exists a point z0 2 E such that

maxjzj�jz0j

jw (z)j = jw (z0)j = 1; w (z0) 6= 1

and also w (z0) = ei� (� 6= ��): Then applying Lemma 2.10.3, we have

z0w0 (z0) = cw (z0) ; c � 1:

68

Using this, we can have

Re

( zf 00(z0)f 0(z0)

zf 0(z0)f(z0)

� 1

)= Re

�1 +

z0w0 (z0)

w (z0) f(1� �)w (z0) + 1g

�= Re

�1 +

cw (z0)

w (z0) f(1� �)w (z0) + 1g

�= 1 + cRe

�1

(1� �)w (z0) + 1

�= 1 + cRe

�1

(1� �) ei� + 1

�= 1 + c

1 + (1� �) cos �

(1� �)2 + 2 (1� �) cos � + 1= F (�) ; say.

Now as we know that F (�) � minF (�) and it can easily be seen that

minF (�) = F (�)

= 1 + c1 + (1� �) cos �

(1� �)2 + 2 (1� �) cos � + 1

= 1 + c1� (1� �)

(1� �)2 � 2 (1� �) + 1

= 1 + c�

(1� � � 1)2

= 1 +c

�

� 1 +1

�

> 1 +1

2� �for � < 1

=3� �

2� �:

Therefore, we have

Re

( zf 00(z0)f 0(z0)

zf 0(z0)f(z0)

� 1

)>3� �

2� �;

69

which is a contradiction to our hypothesis. Thus, we must have jw (z)j < 1 for all z 2 E

and therefore we have from (4:2:1) ;����zf 0 (z)f (z)� 1���� < 1� �;

which shows that zf0(z)f(z)

lies inside a circle centered at (1; 0) and having radius 1�� and we

know from (4:1:5) that this circle lies inside the conic domain k (a; b) ; k 2 [0; 1] ; b � 0

with a and b satisfying (4:1:3) and (4:1:4) : This implies that f (z) 2 k � ST (a; b) ; k 2

[0; 1] ; b � 0 with a and b satisfying (4:1:3) and (4:1:4) :

From the Theorem 4.2.1, we see that when a = 0; b = 0 and k = 1; we have the

following result which is the special case (when p = 1) of the result proved by Al-Kharsani

et.al [6].

Corollary 4.2.1 If f (z) 2 A satis�es the inequality

Re

( zf 00(z)f 0(z)

zf 0(z)f(z)

� 1

)<5

3;

then f (z) is uniformly starlike in E (that is f (z) 2 1� ST ).

The following are the inclusion relations of both classes k�UCV (a; b) and k�ST (a; b)

based on the nested conic regions by varying the parameter b:

Theorem 4.2.2 For b1 > b2;

i. k � UCV (a; b1) � k � UCV (a; b2) :

ii. k � ST (a; b1) � k � ST (a; b2) :

Proof follows directly from Remark 3:1:1(3), (4:1:1) ; (4:1:11) and (4:1:13) :

The following is a radius problem which estimates the radius of k (a; b)-uniform con-

vexity of univalent functions. It gives many known results as special cases, discussed

after the following result.

70

Theorem 4.2.3 Let f (z) 2 S: Then f (z) 2 k � UCV (a; b) for jzj < r0 < 1 with

r0 =2�

p3 + �2

1 + �;

where � is de�ned by (4:1:6) :

Proof. Let f (z) 2 S: Then, for jzj = r < 1; we have����zf 00 (z)f 0 (z)� 2r2

1� r2

���� � 4r

1� r2;

for detail, see [24]. This implies that����(zf 0 (z))0f 0 (z)� 1 + r

2

1� r2

���� � 4r

1� r2: (4.2.3)

The boundary of this disk intersects the real axis at the points�1�4r+r21�r2 ; 0

�and

�1+4r+r2

1�r2 ; 0�:

Now we have to �nd the largest value of r such that the disk (4:2:3) lies completely inside

the conic domain k (a; b) ; that is�1�4r+r21�r2 ; 0

�2 k (a; b) : For this, we must have

1� 4r + r2

1� r2> �;

where � is de�ned by (4:1:6) : This gives us

(1 + �) r2 � 4r + 1� � > 0; 0 < r < 1:

This holds only if

r < r0 =2�

p3 + �2

1 + �:

Now it can also be seen that the curve

(u� a)2 = k2 (u� a+ b� 1)2 + k2v2 + 2k2b (1� b)

71

and for r < r0 =2�p3+�2

1+�; the circle

�u� 1 + r

2

1� r2

�2+ v2 =

16r2

(1� r2)2

do not intersect any where. Therefore, the disk (4:2:3) lies completely inside the conic

domain k (a; b) ; that is, from (4:2:3) we have

(zf 0 (z))0

f 0 (z)2 k (a; b)

which implies that f (z) 2 k � UCV (a; b) : Hence the proof.

When a = 0 and b = 0; then we have the following result, proved by Kanas and

Wisniowska [36].

Corollary 4.2.2 Let f (z) 2 S: Then f (z) 2 k � UCV for jzj < r0 < 1 with

r0 =2 (k + 1)�

p4k2 + 6k + 3

2k + 1:

When a = 0; b = 0 and k = 1; then we have the following result, proved in [96].

Corollary 4.2.3 Let f (z) 2 S: Then f (z) 2 UCV for jzj < r0 < 1 with

r0 =4�

p13

3:

When a = 0; b = 0 and k = 0; then we have the following result, proved in [24].

Corollary 4.2.4 Let f (z) 2 S: Then f (z) 2 C for jzj < r0 < 1 with r0 = 2�p3:

Now we have an extension of the Lemma 2.10.1 proved in [33].

Lemma 4.2.1 Let 0 � k <1: Also, let �; 2 C be such that � 6= 0 and Re (�� + ) >

0; where � is de�ned by (4:1:6) : If p (z) is analytic in E; p (0) = 1; p (z) satis�es

p (z) +zp0 (z)

�p (z) + � pk (a; b; z) ; (4.2.4)

72

and q (z) is analytic solution of

q (z) +zq0 (z)

�q (z) + = pk (a; b; z) ;

then q (z) is univalent, p (z) � q (z) � pk (a; b; z) and q (z) is the best dominant of (4:2:4) :

Proof follows similarly as given in [33].

Moreover, the solution q (z) is given by

q (z) =

��

Z 1

0

�t�+ �1 exp

Z tz

z

pk (a; b;u)� 1u

du

��dt

��1�

�:

For more details of best dominant, we refer to [33, 53].

As a special case, when � = 1 and = 0; we have the function q (z) as

q (z) =

24 1Z0

0@exp tzZz

pk (a; b;u)� 1u

du

1A dt

35�1 : (4.2.5)

Now we see a few applications of the Lemma 4.2.1. When k > 1; the conic domain

k (a; b) may be characterized by the circular domain having its diameter end points as

the vertices of ellipse. As we see that the vertices of ellipse are (�; 0) and (�1; 0) ; where

� is de�ned by (4:1:6) and

�1 = a+k2 (1� b) + k

qk2 (1� b)2 + (1� k2) (1� b2)

k2 � 1 :

The circle K (X;R) having diameter end points (�; 0), (�1; 0) has its center at

X

�a(k2�1)+(1�b)k2

k2�1 ; 0

�and radius R as

R =kqk2 (1� b)2 + (1� k2) (1� b2)

k2 � 1 :

73

The point z = 1 is contained inside the circle K (X;R) and then the function �a;b : E �!

K (X;R) has the form

�a;b (z) = a+k2 (1� b) + kz

p(1� b) (1� b� 2b (k2 � z2))

k2 � z2: (4.2.6)

Theorem 4.2.4 Let k 2 (1;1) and b = 0: Also, let p (z) be analytic in E with p (0) = 1

and p (z) satis�es (4:2:4) : Then

p (z) � 1

(k � z)1R0

ta

k�tzdt

and

Re p (z) >1

(k + 1)1R0

ta

k+tdt

;

where � kk+1

� a < 1k+1

:

Proof. From (4:2:6) ; we have for b = 0;

�a;0 (z) = a+k

k � z:

Since p satis�es (4:2:4) and for each �xed k; pk (a; b; z) � �a;0 (z) ; so p (z) � �a;0 (z) :

This implies from (4:2:5) ;

q (z) =

24 1Z0

0@exp tzZz

�a;0 (u)� 1u

du

1A dt

35�1

=

24 1Z0

0@exp tzZz

a+ kk�u � 1u

du

1A dt

35�1

74

=

24 1Z0

0@exp tzZz

�a

u+

1

k � u

�du

1A dt

35�1

=

24 1Z0

�exp

�log

(tz)a

k � tz� log za

k � z

��dt

35�1

=

24 1Z0

(k � z) ta

k � tzdt

35�1

=1

(k � z)1R0

ta

k�tzdt

:

Also we have

Re p (z) > q (�1) = 1

(k + 1)1R0

ta

k+tdt

for z 2 E:

When a = 0; we have the following result, proved by Kanas [33].

Corollary 4.2.5 Let k 2 (1;1) and let p (z) be analytic in E with p (0) = 1 and p (z)

satis�es (4:2:4) : Then

p (z) � z

(z � k) log�1� z

k

�and

Re p (z) >1

(k + 1) log�1 + 1

k

� :Now we check the integral preservation property of the class k � ST (a; b) under an

integral operator, de�ned in [59]. The technique used in the following result is similar to

that, used by Noor in [59].

Theorem 4.2.5 Let f (z) ; g (z) 2 k � ST (a; b) and let �; c and � be positive reals.

75

Then the function F (z) ; de�ned by

F (z) =

24cz��c zZ0

tc���1 (f (t))� (g (t))��� dt

35 1�

(4.2.7)

belongs to k � ST (a; b) :

Proof. From (4:2:7) ; we have

zc�� (F (z))� = c

zZ0

tc���1 (f (t))� (g (t))��� dt:

This implies that

(c� �) zc���1 (F (z))� + �zc�� (F (z))��1 F 0 (z) = czc���1 (f (z))� (g (z))��� :

Let h (z) = zF 0(z)F (z)

: Then, we have

(F (z))� f(c� �) + �h (z)g = c (f (z))� (g (z))��� :

Di¤erentiating logarithmically, we have

�zF 0 (z)

F (z)+

�zh0 (z)

�h (z) + (c� �)= �

zf 0 (z)

f (z)+ (�� �)

zg0 (z)

g (z):

Now let h1 (z) =zf 0(z)f(z)

and h2 (z) =zg0(z)g(z)

: Then we have

h (z) +zh0 (z)

�h (z) + (c� �)=�

�h1 (z) +

�1� �

�

�h2 (z) :

76

Since f (z) ; g (z) 2 k � ST (a; b) ; so h1 (z) ; h2 (z) 2 k � P (a; b) : And we know by

subordination technique that the class k � P (a; b) is convex. Therefore,

h (z) +zh0 (z)

�h (z) + (c� �)� pk (a; b; z) ;

which implies by using Lemma 4.2.1,

h (z) � pk (a; b; z) :

This shows that F (z) 2 k � ST (a; b) :

When a = 0; b = 0; we get the following result which is a special case of result,

proved by Noor in [59].

Corollary 4.2.6 Let f (z) ; g (z) 2 k � ST and let �; c and � be positive reals. Then

the function F (z) ; de�ned by

F (z) =

24cz��c zZ0

tc���1 (f (t))� (g (t))��� dt

35 1�

(4.2.8)

belongs to k � ST:

4.3 Conclusion

We generalized conic domain k; k � 0 and introduced the conic domain k (a; b) ; k � 0

and discussed various of its aspects. Di¤erent graphical views of this generalized conic

domain for speci�c values of parameters are shown for better understanding of the be-

haviour of this domain. The class k�P (a; b) of functions which map the open unit disk

E onto this generalized conic regions are de�ned and some of its properties are discussed.

Related to the class k � P (a; b) ; two more classes k � UCV (a; b) and k � ST (a; b) are

introduced and some results concerning to these classes are investigated.

77

Chapter 5

On Bounded Boundary and Bounded Radius

Rotation Related with Janowski Function

78

The functions with bounded boundary and bounded radius rotation are discussed in

section 2.7 and the generalized Janowski functions of order � are discussed in section

2.6.2.

In this chapter, we will study the class P [A;B; �] of generalized Janowski functions

of order � along with the functions with bounded boundary and bounded radius rotation.

The order of a function from the class Vm[A;B; �] of Janowski functions with bounded

boundary rotation to be from Rm[A;B; �] of Janowski functions with bounded radius

rotation is our major interest. Some of its applications will also be the part of our

discussion. Our main results present an advancement of already known results.

It is to be mentioned here that all the contents of this chapter have also been published

in "World Applied Sciences Journal, Vol 12(6) 2011, Pages 895 � 902", for detail, see

[79].

5.1 Introduction

The class P [A;B] of Janowski functions is de�ned in De�nition 2.6.9 and its connection

with the class P of functions with positive real part is shown from (2:6:9) : This connection

on the other way can also be seen as:

h (z) 2 P [A;B] () (B � 1)h (z)� (A� 1)(B + 1)h (z)� (A+ 1) 2 P: (5.1.1)

Let P [A;B; �] be the class of functions p1, analytic in E with p1 (0) = 1 and

p1 (z) �1 + [(1� �)A+ �B] z

1 +Bz; �1 � B < A � 1; 0 � � < 1; z 2 E; (5.1.2)

or equivalently,

p1 (z) �1 + Cz

1 +Bz; �1 � B < C � 1; 0 � � < 1; z 2 E;

79

where C = (1� �)A+ �B. It can also be noted that

(1� �) p1 + � 2 P [A;B; �], p1 2 P [A;B] : (5.1.3)

Now we consider the following class, introduced by Noor [56].

De�nition 5.1.1 A function p (z) is said to be in the class Pm[A;B; �], if and only if,

p (z) =

�m

4+1

2

�p1 (z)�

�m

4� 12

�p2 (z) ; (5.1.4)

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

It is clear that P2[A;B; �] � P [A;B; �] and Pm[1;�1; 0] � Pm, the well-known class

given and studied by Pinchuk [87]. The important fact about the class Pm[A;B; �] is

that this class is convex set. That is, for pi (z) 2 Pm[A;B; �] and �i � 0 withnXi=1

�i = 1,

we havenXi=1

�ipi (z) 2 Pm[A;B; �]. (5.1.5)

This can be easily seen from (5:1:3), (5:1:4) and with the fact that the set P [A;B] is

convex [66]. By using all these concepts, we consider the following classes.

Rm[A;B; �] =

�f 2 A : zf

0(z)

f(z)2 Pm [A;B; �] ; z 2 E

�,

Vm[A;B; �] = ff 2 A : zf 0(z) 2 Rm[A;B; �]; z 2 Eg ,

where �1 � B < A � 1, m � 2, and 0 � � < 1. For � = 0 and �1 � B < A � 1,

the classes Vm[A;B; �] and Rm[A;B; �] reduces to the classes Vm[A;B] and Rm[A;B]

respectively, studied by Noor [61, 64, 69].

Throughout this chapter, we assume that C = (1� �)A+ �B unless otherwise men-

tioned. In order to derive our main results, we need the following lemmas.

80

5.2 Preliminary Lemmas

In order to prove our main results, we need the following results.

Lemma 5.2.1 [52] If �1 � B < A � 1; � > 0 and the complex number satis�es

Re f g � �� (1� A) = (1�B) ; (5.2.1)

then the di¤erential equation

q (z) +zq0 (z)

�q (z) + =1 + Az

1 +Bz; z 2 E;

has a univalent solution in E given by

q (z) =

8>>>>>>>>>>><>>>>>>>>>>>:

z�+ (1+Bz)�(A�B)=B

�

zZ0

t�+ �1(1+Bt)�(A�B)=Bdt

� �; B 6= 0;

z�+ e�Az

�

zZ0

t�+ �1e�Atdt

� �; B = 0:

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

h (z) +zh0 (z)

�h (z) + � 1 + Az

1 +Bz; z 2 E; (5.2.2)

then

h (z) � q (z) � 1 + Az

1 +Bz;

and q (z) is the best dominant which can also be written in hypergeometric form as

q (z) =1

�G (z)�

�;

81

where

G(z) =

8>>><>>>:2F1

��(B�A)nB

; 1; �+ +nn

; Bz1+Bz

�(�+ )�1 ; B 6= 0;

1F1�1; �+ +n

n;��

nAz�(�+ )�1 ; B = 0:

Lemma 5.2.2 [110] Let � be a positive measure on [0; 1] and let h (z; t) be a complex-

valued function de�ned on E�[0; 1] such that h (:; t) is analytic in E for each t 2 [0; 1] and

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

0; h (�r; t) is real and

Re

�1

h (z; t)

�� 1

h (�r; t) for jzj � r < 1 and t 2 [0; 1] :

If H (z) =

1Z0

h (z; t) d� (t) ; then Re�

1H(z)

�� 1

H(�r) :

5.3 Main Results

Theorem 5.3.1 Let f (z) 2 Vm [A;B; �] with m � 2; 0 � � < 1; and �1 � B < A � 1

satisfying (5:2:1) : Then, f (z) 2 Rm [A;B; �1] ; where A < �B(1+�)1�� ; B 2 [�1; 0) and

�1 = �1(�; 1; 0) =C

(1�B)B�CB � (1�B)

: (5.3.1)

Proof. Let

zf 0(z)

f(z)= p(z) (5.3.2)

=

�m

4+1

2

�p1 (z)�

�m

4� 12

�p2 (z) : (5.3.3)

Logarithmic di¤erentiation of (5:3:2) yields

(zf 0(z))0

f 0(z)= p(z) +

zp0(z)

p(z).

82

Since f (z) 2 Vm [A;B; �], it follows that

p(z) +zp0(z)

p(z)2 Pm [A;B; �] . (5.3.4)

Now consider a function 'a;b(z) de�ned by Noor [65]

'a;b(z) = z +1Xn=2

b+ 1

b+ (n� 1) a zn

with a = 1; b = 0 and then by using the same convolution technique as used by Noor

[65], we have' (z)

z� p (z) = p(z) +

zp0(z)

p(z): (5.3.5)

From (5:3:3), (5:3:4) and (5:3:5), we obtain

�m

4+1

2

��p1 (z) +

zp01(z)

p1(z)

���m

4� 12

��p2 (z) +

zp02(z)

p2(z)

�2 Pm [A;B; �] : (5.3.6)

From this, we have

pi(z) +zp0i(z)

pi(z)2 P [A;B; �] , i = 1; 2.

We use Lemma 5.2.1 for n = 1, = 0, � = 1 > 0, � 2 [0; 1): and h = pi in (5:2:2), to

have

pi(z) � q (z) � 1 + Cz

1 +Bz: (5.3.7)

This estimate is best possible, extremal function q (z) is given by

q(z) =1

G (z)=

8>>>>><>>>>>:Cz

(1+Bz)�(1+Bz)B�CB

, if B 6= 0,

(1��)Az1�e�(1��)(Az) ; if B = 0.

(5.3.8)

83

From(5:3:7) ; we have

Minjzj=r

Re pi(z) � Minjzj=r

Re q (z) :

Now we show that minRe q(z) = q(�1): Setting a = B�CB; b = 1; c = b + 1 such that

Re c > Re b > 0 and using (2:9:3) ; (2:9:4) and (2:9:5), we have

G (z)= (1 +Bz)a1Z0

tb�1 (1 +Btz)�a dt =� (b)

� (c)2F1

�1; a; c;

Bz

1 +Bz

�; B 6= 0: (5.3.9)

Now we have to show that Re f1=G (z)g � 1=G (�1) ; z 2 E: For A < �B(1+�)1�� ;with

�1 � B < A (Re c > Re a > 0) and using (2:9:3) in (5:3:9), we have

G (z) =

1Z0

g (t; z) d� (t)

where g (z; t) = 1+Bz1+(1�t)Bz and d� (t) =

�(b)�(a) �(c�a)t

a (1� t)c�a�1 dt is a positive measure

on [0; 1] : Now for �1 � B < 0; we have Re g (z; t) > 0 and g (�r; t) is real for 0 � r <

1; t 2 [0; 1] : Also for jzj � r < 1 and t 2 [0; 1] ;

Re

�1

g (z; t)

�= Re

�1 + (1� t)Bz

1 +Bz

�� 1� (1� t)Br

1�Br=

1

g (�r; t) :

Now using the lemma 5.2.2, we obtain Re f1=G (z)g � 1=G (�r) ; (jzj � r < 1) let-

ting r ! 1�; we obtain Re f1=G (z)g � 1=G (�1) : Taking A !��B(1+�)1��

�+and using

(5:3:7) we have consequently from (5:3:8)

�1 = �1(�; 1; 0) = q (�1) = C

(1�B)B�CB � (1�B)

; A < �B (1 + �)1� �

; B 2 [�1; 0) :

This Shows that pi (z) 2 P [A;B; �1], where �1 is given by (5:3:1) and consequently

p (z) 2 Pm [A;B; �1], which gives the required result.

If A = 1; B = �1 in Theorem 5.3.1, we obtain the following result proved in [80].

84

Corollary 5.3.1 Let f (z) 2 Vm(�). Then f (z) 2 Rm(�1), where

�1 = �1(�; 1; 0) =

8><>:2��1

2�22(1��) ; if � 6= 12;

12 ln 2

; if � = 12.

(5.3.10)

If m = 2; A = 1; B = �1 in Theorem 5.3.1, we obtain the following result proved in

[21].

Corollary 5.3.2 Let f (z) 2 C(�). Then f (z) 2 S�(�1), where

�1 = �1(�; 1; 0) =

8><>:2��1

2�22(1��) ; if � 6= 12;

12 ln 2

; if � = 12.

(5.3.11)

Theorem 5.3.2 Let f (z) 2 Vm[A;B; �]: Then f (z) 2 Rm [1;�1; �1], where �1 is one of

the roots of

4 (B2 � 1) �41 + [4 (1� 2�) (B2 � 1)� 8 (1� �) (AB � 1)] �31+

��4�2 � 4� � 3

� �B2 � 1

�+ 4 (1� �)2

�A2 � 1

�� 4 (1� �) (1� 2�) (AB � 1)

��21+

�4 (1� �) (AB � 1)� 2 (1� 2�)

�B2 � 1

���1 +

�B2 � 1

�= 0 (5.3.12)

with 0 � �1 < 1:

Proof. Let

zf 0(z)

f(z)= (1� �1)p(z) + �1 (5.3.13)

= (1� �1)

��m

4+1

2

�p1 (z)�

�m

4� 12

�p2 (z)

�+ �1; (5.3.14)

p(z) is analytic in E with p (0) = 1: Then, di¤erentiating (5:3:13) logarithmically, we

have(zf 0(z))0

f 0(z)= (1� �1)p(z) + �1 +

(1� �1)zp0(z)

(1� �1)p(z) + �1,

85

that is,

1

1� �

�(zf 0(z))0

f 0(z)� �

�=

1

1� �

�(1� �1)p(z) + �1 � � +

(1� �1)zp0(z)

(1� �1)p(z) + �1

�

=(�1 � �)

1� �+(1� �1)

1� �

"p(z) +

1(1��1)

zp0(z)

p(z) + �1(1��1)

#.

Since f (z) 2 Vm[A;B; �]; it implies that

�1 � �

1� �+1� �11� �

"p(z) +

1(1��1)

zp0(z)

p(z) + �1(1��1)

#2 Pm[A;B]; z 2 E. (5.3.15)

Now consider a function 'a;b(z) de�ned by Noor [65]

'a;b(z) = z +1Xn=2

b+ 1

b+ (n� 1) a zn (5.3.16)

with a = 11��1

; b = �11��1

: By using (5:3:16) with the same convolution technique as

used by Noor [65], we have

'a;b(z)

z� p(z) =

�m

4+1

2

��'a;b(z)

z� p1(z)

���m

4� 12

��'a;b(z)

z� p2(z)

�

which implies that

p(z) +azp0(z)

p(z) + b=

�m

4+1

2

��p1(z) +

azp01(z)

p1(z) + b

���m

4� 12

��p2(z) +

azp02(z)

p2(z) + b

�:

(5.3.17)

Thus, from (5:3:15) and (5:3:17), we have

�1 � �

1� �+1� �11� �

�pi(z) +

azp0i(z)

pi(z) + b

�2 P [A;B] ; i = 1; 2: (5.3.18)

86

Using the fact illustrated in (5:1:1), we have

(B � 1)��+ �

hpi(z) +

azp0i(z)pi(z)+b

i�� (A� 1)

(B + 1)��+ �

hpi(z) +

azp0i(z)pi(z)+b

i�� (A+ 1)

2 P;

where � = �1��1�� and � = 1��1

1�� : This implies that

(B � 1) [(�+ �pi(z)) (pi(z) + b) + a�zp0i(z)]� (A� 1) (pi(z) + b)

(B + 1) [(�+ �pi(z)) (pi(z) + b) + a�zp0i(z)]� (A+ 1) (pi(z) + b)2 P:

We now form the functional (u; v) by choosing u = pi(z); v = zp0i(z) and note that the

�rst two conditions of Lemma 2.10.4 are clearly satis�ed. We check condition (iii) as

follows.

(u; v) =(B � 1) [(�+ �u) (u+ b) + a�v]� (A� 1) (u+ b)

(B + 1) [(�+ �u) (u+ b) + a�v]� (A+ 1) (u+ b)

=�1 + a� (B � 1) v + [(�+ � (u+ b)) (B � 1)� (A� 1)]u�2 + a� (B + 1) v + [(�+ � (u+ b)) (B + 1)� (A+ 1)]u ;

where �1 = b [� (B � 1)� (A� 1)] and �2 = b [� (B + 1)� (A+ 1)] : Now

(iu2; v1) =�1 + � (av1 � u22) (B � 1) + [(�+ �b) (B � 1)� (A� 1)] iu2�2 + � (av1 � u22) (B + 1) + [(�+ �b) (B + 1)� (A+ 1)] iu2

:

Taking real part of (iu2; v1), we have

Re (iu2; v1) =

[��1 + � (av1 � u22) (1�B)] [�2 + � (av1 � u22) (B + 1)]�

[(�+ �b) (B � 1)� (A� 1)] [(�+ �b) (B + 1)� (A+ 1)]u22� [�2 + � (av1 + u2) (B + 1)]

2 � [(�+ �b) (B + 1)� (A+ 1)]2 u22:

As a > 0; � > 0, so applying v1 � �12(1 + u22) and after a little simpli�cation, we have

Re (iu2; v1) �A1 +B1u

22 + C1u

42

D1

; (5.3.19)

87

where

A1 =1

4[2�1 � a� (B � 1)] [2�2 � a� (B + 1)] ;

B1 = �12� (a+ 2)

��1 (B + 1)� a�

�B2 � 1

�+ �2 (B � 1)

�+

(�+ �b)2�B2 � 1

�� 2 (�+ �b) (AB � 1) +

�A2 � 1

�;

C1 = �14�2�1�B2

�(a+ 2)2 ;

and

D1 = [�2 + � (av1 + u2) (B + 1)]2 + [(�+ �b) (B + 1)� (A+ 1)]2 u22:

The right hand side of (5:3:19) is negative if A1 � 0 and B1 � 0: From A1 � 0, we have

�1 to be one of the roots of

4�B2 � 1

��41 +

�4 (1� 2�)

�B2 � 1

�� 8 (1� �) (AB � 1)

��31+��

4�2 � 4� � 3� �B2 � 1

�+ 4 (1� �)2

�A2 � 1

�� 4 (1� �) (1� 2�) (AB � 1)

��21+�

4 (1� �) (AB � 1)� 2 (1� 2�)�B2 � 1

���1 +

�B2 � 1

�= 0

with 0 � �1 < 1 and also for 0 � �1 < 1, we have

B1 = 4�B2 � 1

��31 + 4

�(1 + �)

�B2 � 1

�+ (1� �) (AB � 1)

��21���

2�2 + 2� + 5� �B2 � 1

�+ 2 (1 + 2�) (1� �) (AB � 1) + 2

�A2 � 1

�(1� �)2

��1+��

3 + 2�2� �B2 � 1

�+ 4� (1� �) (AB � 1) + 2

�A2 � 1

�(1� �)2

�� 0:

Since all the conditions of Lemma 2.10.4 are satis�ed, it follows that pi (z) 2 P in E for

i = 1; 2 and consequently p (z) 2 Pm [1;�1] and hence f (z) 2 Rm [1;�1; �1], where �1 is

one of the roots of (5:3:12) with 0 � �1 < 1:

88

By setting A = 1; B = �1 in Theorem 5.3.2, we obtain the following result proved in

[80].

Corollary 5.3.3 Let f (z) 2 Vm (�) : Then f (z) 2 Rm (�1), where �1 is one of the roots

of

2�21 � (2� � 1) �1 � 1 = 0 with 0 � �1 < 1;

which is

�1 =1

4

�(2� � 1) +

q4�2 � 4� + 9

�:

If m = 2; A = 1; B = �1 in Theorem 5.3.2, we obtain the following result proved in

[21].

Corollary 5.3.4 Let f (z) 2 C (�) : Then f (z) 2 S� (�1), where �1 is one of the roots

of

2�21 � (2� � 1) �1 � 1 = 0 with 0 � �1 < 1;

which is

�1 =1

4

�(2� � 1) +

q4�2 � 4� + 9

�:

Application of Theorem 5.3.1

Theorem 5.3.3 Let f (z) and g (z) belong to Vm [A;B; �] with m � 2; 0 � � < 1; and

�1 � B < A � 1: Then the function F (z) de�ned by

(F (z))�1 =c

zc��1

zZ0

t(c����)�1 (f (t))� (g (t))� dt; �1; c; �; � > 0; (5.3.20)

is in the class Rm [A;B; �1] ; where

�1 =c

�1 2F1�1; �1(1� �)

�1� A

B

�; c+ 1; B

B�1� � c� �1

�1(5.3.21)

and c > �1; A < � [(c+1)��1(1��)]B�1(1��) ; (� + �) = �1:

89

Proof. Let

(zF 0(z))0

F 0(z)= p (z) (5.3.22)

=

�m

4+1

2

�p1 (z)�

�m

4� 12

�p2 (z) : (5.3.23)

Then, p (z) is analytic in E and p (0) = 1: With zf 0(z)f(z)

= p1 (z) ;zg0(z)g(z)

= p2 (z) ; we have

from (5:3:20) and by using (5:3:22)

p (z) +zp0(z)

�1p(z) + (c� �1)=

�

�1p1 (z) +

�

�1p2 (z) = H0 (z) :

Since f (z) ; g (z) 2 Vm [A;B; �] and this means that f (z) ; g (z) 2 Rm [A;B; �1] ; so

p1 (z) ; p2 (z) 2 Pm [A;B; �1] : It is known that Pm [A;B; �1] is convex set: Therefore,

H0 (z) 2 Pm [A;B; �1] and �1 = �1 (�) is given by (5:3:1) : This implies that

p (z) +zp0(z)

�1p(z) + (c� �1)2 Pm [A;B; �1] : (5.3.24)

Now consider a function 'a;b(z) de�ned by Noor [65]

'a;b(z) = z +1Xn=2

b+ 1

b+ (n� 1) a zn

with a = �1; b =c��1�1

and using (5:3:23) with the same convolution technique as used

by Noor [65], we have

'a;b(z)

z� p (z) =

�m

4+1

2

��'a;b(z)

z� p1 (z)

���m

4� 12

��'a;b(z)

z� p2 (z)

�;

90

which implies that

p(z) +zp0(z)

�1p(z) + (c� �1)=

�m

4+1

2

��p1 (z) +

zp01(z)

�1p1(z) + (c� �1)

���m

4� 12

��p2 (z) +

zp02(z)

�1p2(z) + (c� �1)

�:(5.3.25)

Thus, from (5:3:24) and (5:3:25), we have

pi (z) +zp0i(z)

�1pi(z) + (c� �1)2 P [A;B; �1] , i = 1; 2.

Therefore,

pi (z) +zp0i(z)

�1pi(z) + (c� �1)� 1 + f(1� �1)A+ �1Bg z

1 +Bz:

Using Lemma 5.2.1 for � = �1; = c� �1; we have

pi (z) � q (z) � 1 + f(1� �1)A+ �1Bg z1 +Bz

; (5.3.26)

where

q (z) =c

�1G (z)� c� �1

�1;

and

G(z) =

8><>: 2F1

�1; �1(B�C)

B; c+ 1; Bz

1+Bz

�; if B 6= 0;

1F1 (1; c+ 1;��1Az) ; if B = 0:

From (5:3:26) ; we have

Minjzj=r

Re pi(z) � Minjzj=r

Re q (z) :

By using the Lemma 5.2.2, we have minRe q(z) = q(�1): Consequently, we have p (z) 2

Pm [A;B; �1] where �1 is given by (5:3:21) : This shows that F (z) 2 Rm [A;B; �1].

Theorem 5.3.4 Let f (z) and g (z) belong to Vm [A;B; �] with m � 2; 0 � � < 1; and

�1 � B < A � 1: Then, the function F (z) with �1 = c = 1; de�ned by (5:3:20) is in the

91

class Vm [A;B; �] ; where 0 � � < � � 1;

� = �(�) = (1� (� + �) (1� �1)) (5.3.27)

and �1(�) is given by (5:3:1) :

Proof. From (5:3:20) ; we can easily write

(zF 0(z))0

F 0(z)= �

zf 0(z)

f(z)+ �

zg0(z)

g(z)+ 1� (� + �): (5.3.28)

Since f (z) and g (z) belong to Vm [A;B; �], then, by Theorem 5.3.1, zf 0(z)f(z)

and zg0(z)g(z)

belong to Pm [A;B; �1], where �1 = �1(�) is given by (5:3:1) : Using

zf 0(z)

f(z)= (1� �1)p1(z) + �1, p1 (z) 2 Pm [A;B] ;

andzg0(z)

g(z)= (1� �1)p2(z) + �1, p2 (z) 2 Pm [A;B] ;

in (5:3:28) ; we have

1

1� �

�(zF 0(z))0

F 0(z)� �

�=

�

� + �p1(z) +

�

� + �p2(z): (5.3.29)

Now by using the well known fact that the class Pm [A;B] is a convex set together with

(5:3:29), we obtain the required result.

5.4 Conclusion

We studied the class P [A;B; �] of generalized Janowski functions of order � along with

the functions with bounded boundary and bounded radius rotation. We investigated the

inclusion result with regard to order from the class Vm[A;B; �] to the class Rm[A;B; �]

in two di¤erent prospectives and some of its applications are also discussed.

92

Chapter 6

On Generalization of a Class of Analytic Functions

De�ned by Ruscheweyh Derivative

93

In 1994, Latha and Nanjunda [39] introduced the class Vm (�; b; �) of analytic functions

by using the Ruscheweyh derivative. As we discussed in Chapter 2 and 3 that the classes

of convex and starlike functions can be obtained from the class of �-convex functions

as special cases. This can also be done by Vm (�; b; �) ; that is, for speci�c values of the

parameters m; � and b; the class Vm (�; b; �) gives convex and starlike functions.

In this chapter, we will generalize the class Vm (�; b; �) with the concept of Janowski

functions and introduce the class V�m [A;B; �; b] : This generalized class contains many

known classes. The coe¢ cient bound, inclusion result and a radius problem will be

investigated here. Several known results will also be deducted from our main results as

special cases by assigning particular values to di¤erent parameters.

6.1 Introduction

Using the concept of generalized Janowski functions of order �; denoted by the class

P [A;B; �] ; discussed in section 2.6.2, we generalize the class Vm (�; b; �) ; introduced

by Latha and Nanjunda [39] but �rst we recall the de�nition of Pm[A;B; �]; given by

De�nition 5.1.1, for �1 � B < A � 1:

De�nition 6.1.1 A function p (z) is said to be in the class Pm[A;B; �]; if and only if;

p (z) =

�m

4+1

2

�p1 (z)�

�m

4� 12

�p2 (z) ; (6.1.1)

where p1 (z) ; p2 (z) 2 P [A;B; �] ; �1 � B < A � 1; m � 2; and 0 � � < 1.

It is clear that P2[A;B; �] � P [A;B; �] and Pm[1;�1; 0] � Pm; the well-known class

given and studied by Pinchuk [87].

Now using the concepts Ruscheweyh derivative along with the above class, we de�ne

the following.

94

De�nition 6.1.2 A function f (z) 2 A is in the class V�m [A;B; �; b] ; if and only if;�1� 2

b+2

b

D�+1f(z)

D�f(z)

�2 Pm [A;B; �] ; z 2 E;

where m � 2; � > �1; �1 � B < A � 1; 0 � � < 1 and b 2 C� f0g.

Assigning certain values to di¤erent parameters, we have di¤erent well-known classes

of analytic functions as can be seen below.

Special Cases

(i) V�m [1;�1; �; b] � Vm (�; b; �) ; the well-known class de�ned by Latha and Nanjunda

Rao in [39].

(ii) V12 [A;B; �; 1] � C [A;B; �] ; V02 [A;B; �; 2] � S� [A;B; �] ; the well-known class

de�ned by Polato¼glu [89].

(iii) V1m [A;B; 0; 1] � Vm [A;B] ; V02 [A;B; 0; 2] � Rm [A;B] ; where Vm [A;B] andRm [A;B]

denote the classes of Janowski functions with bounded boundary and bounded ra-

dius rotations respectively; given and studied by Noor [61, 64, 69].

6.2 Preliminary Results

We need the following results to obtain our main results.

Lemma 6.2.1 Let p (z) = 1 +1Pn=1

qnzn 2 Pm [A;B; �] : Then; for all n � 1;

jqnj �m (A�B) (1� �)

2: (6.2.1)

This inequality is sharp.

The proof follows from (2.4.1) ; (6.1.1) and the coe¢ cient bound of h 2 P [A;B] given

by Aouf [8]:

95

Lemma 6.2.2 Let p (z) 2 Pm [A;B; 0] with m � 2: Then; for jzj = r < 1;

2� (A�B)mr � 2ABr22 (1�B2r2)

� Rep(z) � jp(z)j � 2 + (A�B)mr � 2ABr22 (1�B2r2)

: (6.2.2)

The proof is immediate by using (6.1.1) and the growth result of h 2 P [A;B] ; see

[86].

Lemma 6.2.3 Let p (z) 2 Pm [A;B; 0] with m � 2: Then; for jzj = r < 1;

jzp0(z)j � r f(A�B)m� 4B (A�B) r +B2 (A�B)mr2gRep(z)(1�B2r2) (2 + (A�B)mr � 2ABr2) : (6.2.3)

The proof follows directly by using Lemma 6.2.2.

6.3 Main Results

Theorem 6.3.1 Let f (z) 2 V�m [A;B; �; b] with �1 � B < A � 1; � > �1; b 2

C� f0g ; 0 � � < 1: Then

janj �(�)n�1

(n� 1)!'n (�); n � 2; (6.3.1)

where � =m jbj (A�B) (1� �) (� + 1)

4and 'n (�) is given by (2.8.2) :

This result is sharp.

Proof. Set

1� 2b+2

b

D�+1f(z)

D�f(z)= p (z) ; (6.3.2)

so that p (z) 2 Pm [A;B; �] : Let p (z) = 1 +1Pn=1

qn zn: Then (6.3.2) can be written as

2�D�+1f(z)�D�f(z)

�= bD�f(z)

1Xn=1

qn zn;

96

which implies that

21Xn=2

('n (� + 1)� 'n (�)) anzn = b

1Xn=1

'n (�) anzn

! 1Xn=1

qn zn

!:

This implies that

21Xn=2

�� + n

� + 1'n (�)� 'n (�)

�anz

n =1Xn=2

bn�1Xj=1

'j (�) aj qn�j

!zn; a1 = 1:

Equating coe¢ cients of zn; we have

2'n (�) (n� 1) an(� + 1)

= bn�1Xj=1

'j (�) aj qn�j:

Using Lemma 6.2.1; we obtain

janj �m jbj (A�B) (1� �) (� + 1)

4 (n� 1)'n (�)

n�1Xj=1

'j (�) jajj

=�

(n� 1)'n (�)

n�1Xj=1

'j (�) jajj ; (6.3.3)

where � =m jbj (A�B) (1� �) (� + 1)

4: Now we prove that

n�1Xj=1

'j (�) jajj �n�2Yj=1

�1 +

�

j

�: (6.3.4)

For this, we use induction method. We see that it is true for n = 2; 3: Let it be true for

n = t; that is,t�1Xj=1

'j (�) jajj �t�2Yj=1

�1 +

�

j

�(6.3.5)

97

which gives along with (6:3:3)

jatj ��

(t� 1)'t (�)

t�2Yj=1

�1 +

�

j

�: (6.3.6)

Then considertXj=1

'j (�) jajj =t�1Xj=1

'j (�) jajj+ 't (�) jatj :

Using (6:3:5) and (6:3:6) in above relation, we have

tXj=1

'j (�) jajj �t�2Yj=1

�1 +

�

j

�+ 't (�)

�

(t� 1)'t (�)

t�2Yj=1

�1 +

�

j

�

=

�1 +

�

t� 1

� t�2Yj=1

�1 +

�

j

�

=t�1Yj=1

�1 +

�

j

�;

which shows that (6:3:4) is true for all n � 2: Thus from (6:3:3) and (6:3:4) ; we have

janj ��

(n� 1)'n (�)

n�2Yj=1

�1 +

�

j

�

=�

(n� 1)'n (�)

n�2Qj=1

(� + j)

n�2Qj=1

j

=(�)n�1

(n� 1)!'n (�):

98

This result is sharp for � > �1; 0 � � < 1; b 2 C� f0g and m � 2 as can be seen from

the functions f0 (z) which are given as

1� 2b+2

b

D�+1f0 (z)

D�f0 (z)= (1� �)

��m

4+1

2

�1 + Az

1 +Bz��m

4� 12

�1� Az

1�Bz

�+ �:

For di¤erent values of A; B; �; b and �; we obtain the following known results, proved

by Noor [57].

Corollary 6.3.1 If f (z) 2 V0m [1;�1; �; 2] = Rm (�) ; then

janj �(m (1� �))n�1(n� 1)! ; 8 n � 2:

This result is sharp.

Corollary 6.3.2 If f (z) 2 V1m [1;�1; �; 1] = Vm (�) ; then

janj �(m (1� �))n�1

n!; 8 n � 2:

This result is sharp.

Theorem 6.3.2 For real b > 0; V�+1m [A;B; �; b] � V�m [1;�1; �2; b+ 1] ; z 2 E; where

�2 (0 � �2 < 1) is one of the roots of

�1�2b2 (� + 2)2 (1� �)2 � b (� + 2) (1� �) [�1 (B + 1) + �2 (B � 1)] +

�B2 � 1

�= 0;

(6.3.7)

where

�1 =(1� b) + � (1 + b)

(1 + b) (1� �)

��1� (1 + b) (� + 1) (1� �)

b (� + 2) (1� �)

�(B � 1)� (A� 1)

�(6.3.8)

99

and

�2 =(1� b) + � (1 + b)

(1 + b) (1� �)

��1� (1 + b) (� + 1) (1� �)

b (� + 2) (1� �)

�(B + 1)� (A+ 1)

�: (6.3.9)

Proof. Suppose f (z) 2 V�+1m [A;B; �; b] and set

p (z) = 1� 2

b+ 1+

2

b+ 1

D�+1f(z)

D�f(z): (6.3.10)

where p (z) is analytic in E with p (0) = 1: Then

p (z) +2

b+ 1� 1 = 2

b+ 1

D�+1f(z)

D�f(z):

Di¤erentiating logarithmically, we have

zp0 (z)

p (z) + 2b+1

� 1=z�D�+1f(z)

�0D�+1f(z)

�z�D�f(z)

�0D�f(z)

which implies by using (2.8.4) that

zp0 (z)

p (z) + 2b+1

� 1=

(� + 2)D�+2f(z)� (� + 1)D�+1f(z)

D�+1f(z)� (� + 1)D

�+1f(z)� �D�f(z)

D�f(z)

=(� + 2)D�+2f(z)

D�+1f(z)� 1� (� + 1)D

�+1f(z)

D�f(z):

This implies that

1� 2b+2

b

D�+2f(z)

D�+1f(z)= 1� 2

b+

2

b (� + 2)+2 (� + 1)

b (� + 2)

D�+1f(z)

D�f(z)

+2

b (� + 2)

zp0 (z)

p (z) + 2b+1

� 1;

100

which reduces to

1� 2b+2

b

D�+2f(z)

D�+1f(z)= 1� b+ 1

b

� + 1

� + 2+b+ 1

b

� + 1

� + 2

�1� 2

b+ 1+

2

b+ 1

D�+1f(z)

D�f(z)

+2

(b+ 1) (� + 1)

zp0 (z)

p (z) + 2b+1

� 1

#

= (1� �1) + �1

�p(z) +

�2 zp0 (z)

p (z) + �3

�; (6.3.11)

where �1 =�+1�+2

b+1b; �2 =

2(�+1)(b+1)

; �3 =2b+1�1: Since f (z) 2 V�+1m [A;B; �; b] ; it follows

that

(1� �1) + �1

�p(z) +

�2 zp0 (z)

p (z) + �3

�2 Pm [A;B; �] ;

or, equivalently

(1� � � �1)

1� �+

�11� �

�p(z) +

�2 zp0 (z)

p (z) + �3

�2 Pm [A;B] : (6.3.12)

De�ne

' (z) =1

(1 + �3)

z

(1� z)�2+

�3(1 + �3)

z

(1� z)�2+1;

and by using convolution technique given by Noor [56], we have

p(z) +�2 zp

0 (z)

p (z) + �3=

�m

4+1

2

��p1 (z) +

�2 zp01 (z)

p1 (z) + �3

�

��m

4� 12

��p2 (z) +

�2 zp02 (z)

p2 (z) + �3

�:

By using (6.3.12) ; we see that

(1� � � �1)

1� �+

�11� �

�pi (z) +

�2 zp0i (z)

pi (z) + �3

�2 P [A;B] ; z 2 E; i = 1; 2:

101

Now; we want to show that pi (z) 2 P [A;B; �2] ; where �2 (0 � �2 < 1) is one of the

roots of (6.3.7). Let

pi (z) = (1� �2)hi (z) + �2; i = 1; 2:

Then

1� � � �1 (1� �2)

1� �+�1 (1� �2)

1� �

"hi (z) +

�2(1��2)

zh0i (z)

hi (z) +�3+�2(1��2)

#2 P [A;B] :

Using the fact illustrated in (5.1.1) ; we have

(B � 1)h�+ �

�hi (z) +

!1 zh0i(z)hi(z)+!2

�i� (A� 1)

(B + 1)h�+ �

�hi (z) +

!1 zh0i(z)hi(z)+!2

�i� (A+ 1)

2 P

where !1 =�21��2

; !2 =�3+�21��2

; � = 1����1(1��2)1�� and � = �1(1��2)

1�� : This implies that

(B � 1) [(�+ �hi(z)) (hi(z) + !2) + !1�zh0i(z)]� (A� 1) (hi(z) + !2)

(B + 1) [(�+ �hi(z)) (hi(z) + !2) + !1�zh0i(z)]� (A+ 1) (hi(z) + !2)2 P;

We now form the functional (u; v) by choosing u = hi(z); v = zh0i(z) and note that the

�rst two conditions of Lemma 2.10.4 are clearly satis�ed. We check condition (iii) as

follows.

(u; v) =(B � 1) [(�+ �u) (u+ !2) + !1�v]� (A� 1) (u+ !2)

(B + 1) [(�+ �u) (u+ !2) + !1�v]� (A+ 1) (u+ !2)

=�1 + !1� (B � 1) v + [(�+ � (u+ !2)) (B � 1)� (A� 1)]u�2 + !1� (B + 1) v + [(�+ � (u+ !2)) (B + 1)� (A+ 1)]u

:

where �1 = !2 [� (B � 1)� (A� 1)] and �2 = !2 [� (B + 1)� (A+ 1)] : Now

(iu2; v1) =�1 + � (!1v1 � u22) (B � 1) + [(�+ �!2) (B � 1)� (A� 1)] iu2�2 + � (!1v1 � u22) (B + 1) + [(�+ �!2) (B + 1)� (A+ 1)] iu2

:

Taking real part of (iu2; v1); we have

102

Re (iu2; v1) =

[��1 + � (!1v1 � u22) (1�B)] [�2 + � (!1v1 � u22) (B + 1)]�

[(�+ �!2) (B � 1)� (A� 1)] [(�+ �!2) (B + 1)� (A+ 1)]u22�[�2+�(!1v1+u2)(B+1)]2�[(�+�!2)(B+1)�(A+1)]2u22

:

As !1 > 0; � > 0; so applying v1 � �12(1 + u22) and after a little simpli�cation; we have

Re (iu2; v1) �A1 +B1u

22 + C1u

42

D1

; (6.3.13)

where

A1 =1

4[2�1 � !1� (B � 1)] [2�2 � !1� (B + 1)] ;

B1 = �12� (!1 + 2)

��1 (B + 1)� !1�

�B2 � 1

�+ �2 (B � 1)

�+

(�+ �!2)2 �B2 � 1

�� 2 (�+ �!2) (AB � 1) +

�A2 � 1

�;

C1 = �14�2�1�B2

�(!1 + 2)

2 ;

and

D1 = [�2 + � (!1v1 + u2) (B + 1)]2 + [(�+ �!2) (B + 1)� (A+ 1)]2 u22:

The right hand side of (6.3.13) is negative if A1 � 0 and B1 � 0: From A1 � 0; we have

�2 to be one of the roots of

�1�2b2 (� + 2)2 (1� �)2 � b (� + 2) (1� �) [�1 (B + 1) + �2 (B � 1)] +

�B2 � 1

�= 0

with 0 � �2 < 1 and also for 0 � �2 < 1; we have B1 � 0:

Since all the conditions of Lemma 2.10.4 are satis�ed; it follows that hi (z) 2 P; i = 1; 2

and consequently p (z) 2 Pm [1;�1; �2] :Hence from (6.3.10) ; f (z) 2 V�m [1;�1; �2; b+ 1] :

103

By choosing the parameters A = 1; B = �1; b = 1 and � = 0; we obtain the following

known result; proved in [80].

Corollary 6.3.3 Let f (z) 2 Vm (�) : Then f (z) 2 Rm (�2) ; where �2 is a root of

2�22 � (2� � 1) �2 � 1 = 0 with 0 � �2 < 1;

which is

�2 =1

4

�(2� � 1) +

q4�2 � 4� + 9

�:

Corollary 6.3.4 For � = 0; m = 2 in Corollary 6.3.3; we have the following well known

result [24].

V2 (0) = C � R2

�1

2

�= S�

�1

2

�; for z 2 E.

Theorem 6.3.3 Let f (z) 2 V�m [A;B; 0; b] ; � > �1; b > 0 (real); m � 2 and 0 < a =

b(�+1)2

� 1. Then D�f(z) maps jzj < r0 onto a convex domain; where r0 is the least

positive root of the equation

a1r4 + a2r

3 + a3r2 + a4r + 4 (2a� 1) = 0 with 0 � r < 1; (6.3.14)

where

a1 = 4a2A2B2 � 4 (a� 1)2B4;

a2 = 2a (2a� 1) (B � A)B2m;

a3 = 8a2 (a� 2) + 8a (1� a)AB � a2 (A�B)2m2;

and

a4 = 2a (2a� 3) (A�B)m:

This result is sharp.

104

Proof. Since f (z) 2 V�m [A;B; 0; b] ; then

D�+1f(z)

D�f(z)=b (p (z)� 1) + 2

2; (6.3.15)

where p (z) 2 Pm [A;B; 0] : Using the identity (2.8.4) ; we consider

z�D�f(z)

�0D�f(z)

= (� + 1)D�+1f(z)

D�f(z)� �:

Using (6.3.15) in above relation, we obtain

z�D�f(z)

�0D�f(z)

=b (p (z)� 1) (� + 1) + 2

2: (6.3.16)

Logarithmic di¤erentiation of (6.3.16) yields

�z�D�f(z)

�0�0(D�f(z))0

�z�D�f(z)

�0D�f(z)

=b(�+1)2

zp0 (z)b(�+1)2

p (z)� b(�+1)2

+ 1

which implies that

�z�D�f(z)

�0�0(D�f(z))0

= ap(z)� a+ 1 +zp0(z)

p(z)� 1 + 1a

;

where a = b(�+1)2

: Then; we have

Re

1 +

z�D�f(z)

�00(D�f(z))0

!� aRe p(z) + (1� a)� a jzp0(z)j

jap(z) + 1� aj : (6.3.17)

105

Now using Lemma 6.2.2, we see that

1

jap(z) + 1� aj � 1

a jp(z)j � j1� aj

� 1

a2�(A�B)mr�2ABr2

2(1�B2r2) � (1� a); a � 1

=2 (1�B2r2)

2 (2a� 1)� a (A�B)mr + 2 (B2 � a (A+B)B) r2:(6.3.18)

Using (6:3:18) ; Lemma 6.2.2 and Lemma 6.2.3; we have from (6:3:17) ;

Re

1 +

z�D�f(z)

�00(D�f(z))0

!� Re p(z)

�a+

2 (1� a) (1�B2r2)

2 + (A�B)mr � 2ABr2

� 2ar f(A�B)m� 4B (A�B) r +B2 (A�B)mr2g(2 + (A�B)mr � 2ABr2) �

�

= Re p(z)

�a1r

4 + a2r3 + a3r

2 + a4r + 4 (2a� 1)(2 + (A�B)mr � 2ABr2) �

�> 0;

provided

T (r) = a1r4 + a2r

3 + a3r2 + a4r + 4 (2a� 1) > 0;

where

a1 = 4a2A2B2 � 4 (a� 1)2B4;

a2 = 2a (2a� 1) (B � A)B2m;

a3 = 8a2 (a� 2) + 8a (1� a)AB � a2 (A�B)2m2;

a4 = 2a (2a� 3) (A�B)m;

106

and

� = 2 (2a� 1)� a (A�B)mr + 2�B2 � a (A+B)B

�r2:

We have T (0) > 0 and T (1) < 0: Therefore D�f(z) maps jzj < r0 onto a convex domain;

where r0 is the least positive root of the equation T (r) = 0; lying in (0; 1).

For D�f1 (z) such thatD�+1f1 (z)

D�f1 (z)=b (pm (z)� 1) + 2

2;

where pm (z) =2+(A�B)mz�2ABz2

2(1�B2z2) ; we have

�z�D�f1(z)

�0�0(D�f1(z))

0 =a1r

4 + a2r3 + a3r

2 + a4r + 4 (2a� 1)(2 + (A�B)mr � 2ABr2) � = 0;

for z = r0. Hence this radius r0 is sharp.

By choosing the parameters A = 1; B = �1; m = 2; b = 2 and � = 0; we obtain the

following known result [24].

Corollary 6.3.5 Let f (z) 2 S�. Then f (z) maps jzj < r0 onto a convex domain; where

r0 is the least positive root of the equation

r4 � 2r3 � 6r2 � 2r + 1 = 0 with 0 � r < 1;

which is r0 = 2�p3: This is also sharp.

6.4 Conclusion

We generalized the class Vm (�; b; �) with the concept of Janowski functions and intro-

duced the class V�m [A;B; �; b] : This generalized class contains many known classes and

we included them as special cases. The coe¢ cient bound, inclusion result and a radius

problem are investigated. Several known results are deducted from our main results as

special cases by assigning particular values to di¤erent parameters.

107

Chapter 7

On Janowski Functions Associated with Conic

Domains

108

In the study of analytic functions, geometry of image domain has undoubtedly a great

importance. Analytic functions are classi�ed into many classes and then into subclasses

depending upon the shape of image domain and other geometrical properties. It has been

a matter of discussion that there always exist analytic functions with di¤erent geometrical

structures as their image domains. Since Goodman [22, 23] introduced parabolic region as

image domain and de�ned the function which gives exactly parabolic region as its image

domain. After that Kanas and Wisniowska [36, 35] contributed in form of hyperbolic and

elliptic regions along with their extremal functions. For details, see section 2.6.1. Before

all this, Janowski introduced circular regions and their extremal functions as discussed

in section 2.6.2.

In this chapter, we will introduce new geometrical structures of oval and petal type

shape as image domain and de�ne the classes of functions which give these types of map-

pings. The concepts of Janowski functions and conic domains are combined together to

de�ne a new domain k [A;B] which represents the oval and petal type regions. Di¤erent

graphical views of this new domain for speci�c values of parameters will be shown in or-

der to have better understanding of the behaviour of this domain k [A;B] : This domain

gives both conic and circular domains as special cases which is the main motivation of

this chapter. Moreover, two new classes k�UCV [A;B] of k-uniformly Janowski convex

and k � ST [A;B] of k- Janowski starlike functions will be de�ned and our main results

will be based on their properties.

A number of already known classes of analytic functions can easily be obtained from

our new classes as special cases. The class SD (k; �) discussed in section 2.6.1 is also a

special case of our new class k�ST [A;B] : The coe¢ cient bound for the class SD (k; �),

proved by Owa et al. [83] is improved, that is, the coe¢ cient bound for class SD (k; �)

obtained from our main results as special case gives much better result as compared to

that one, proved by Owa et al. [83].

It is to be mentioned here that some of the contents of this chapter have been pub-

lished in a well reputed journal "Computers and Mathematics with Applications, Vol

109

62; 2011; Pages 2209� 2217", for detail, see[73].

7.1 Introduction

The classes S� (�) and C (�) are the well-known classes of starlike and convex univalent

functions of order � (0 � � < 1) respectively, discussed in section 2.4. The class P [A;B]

represents the class of Janowski functions which we have de�ned in section 2.6.2 and its

connection with the class P of functions with positive real part is also shown there by

(2:6:9) : The circular domain [A;B] from which Janowski functions take values is given

by (2:6:8) :

Kanas and Wisniowska [36, 35] introduced and studied the class k � UCV of k-

uniformly convex functions and the corresponding class k � ST of k-starlike functions.

These classes were de�ned subject to the conic domain k; k � 0 which is de�ned by

(2:6:1) : These classes were then generalized to KD (k; �) and SD (k; �) respectively by

Shams et.al [101] subject to the conic domain G (k; �) ; k � 0; 0 � � < 1; which is

de�ned by (2:6:7) :

Now using the concepts of Janowski functions and the conic domain, we de�ne the

following.

De�nition 7.1.1 A function p (z) is said to be in the class k � P [A;B] ; if and only if,

p (z) � (A+ 1) pk(z)� (A� 1)(B + 1) pk(z)� (B � 1)

; k � 0; (7.1.1)

where pk(z) is de�ned by (2:6:2) and �1 � B < A � 1: Geometrically, the function

p (z) 2 k�P [A;B] takes all values from the domain k [A;B] ; �1 � B < A � 1; k � 0

which is de�ned as:

k [A;B] =

�w : Re

�(B � 1)w(z)� (A� 1)(B + 1)w(z)� (A+ 1)

�> k

����(B � 1)w(z)� (A� 1)(B + 1)w(z)� (A+ 1) � 1�����(7.1.2)

110

or equivalently

k [A;B] =nu+ iv :

��B2 � 1

� �u2 + v2

�� 2 (AB � 1)u+

�A2 � 1

��2> k2

h��2 (B + 1)

�u2 + v2

�+ 2 (A+B + 2)u� 2 (A+ 1)

�2+4 (A�B)2 v2

�:

The domain k [A;B] represents the conic type regions as shown in the �gures below.

Figure 7.1: Boundary of domain k [0:5;�0:5]

Figure 7.2: Boundary of domain k [0:8; 0:2]

111

The domain k [A;B] retains the conic domain k inside the circular region de�ned by

[A;B] : The impact of [A;B] on the conic domain k changes the original shape of

the conic regions. The ends of hyperbola and parabola get closer to each other but never

meet anywhere and the ellipse gets the shape of oval as shown in �gures 7.1, 7.2 above.

When A �! 1; B �! �1; the radius of the circular disk de�ned by [A;B] tends to

in�nity, consequently the arms of hyperbola and parabola expand and the oval turns into

ellipse as shown in �gures below.

Figure 7.3: View of k [A;B] when A! 1; B ! �1:

Figure 7.4: Close view of Fig. 7.3.

It can be seen that k [1;�1] = k; the conic domain de�ned by Kanas and Wisniowska

[36]. Here are some basic facts about the class k � P [A;B] :

112

Remark 7.1.1 1. k � P [A;B] � P�2k+1�A2k+1�B

�; the class of functions with real part

greater than 2k+1�A2k+1�B :

2. k � P [1;�1] = P (pk) ; the well-known class introduced by Kanas and Wisniowska

[36].

3. 0� P [A;B] = P [A;B] ; the well-known class introduced by Janowski [30].

Now we de�ne the classes k � UCV [A;B] of k-uniformly Janowski functions and

k � ST [A;B] of k-Janowski starlike functions as follows.

De�nition 7.1.2 A function f (z) 2 A is said to be in the class k�UCV [A;B] ; k � 0;

�1 � B < A � 1; if and only if ,

Re

0@(B � 1) (zf 0(z))0f 0(z) � (A� 1)

(B + 1) (zf0(z))0

f 0(z) � (A+ 1)

1A > k

������(B � 1)(zf 0(z))0

f 0(z) � (A� 1)

(B + 1) (zf0(z))0

f 0(z) � (A+ 1)� 1

������ ;or equivalently,

(zf 0(z))0

f 0 (z)2 k � P [A;B] : (7.1.3)

De�nition 7.1.3 A function f (z) 2 A is said to be in the class k � ST [A;B] ; k � 0;

�1 � B < A � 1; if and only if ,

Re

(B � 1) zf

0(z)f(z)

� (A� 1)

(B + 1) zf0(z)f(z)

� (A+ 1)

!> k

�����(B � 1)zf 0(z)f(z)

� (A� 1)

(B + 1) zf0(z)f(z)

� (A+ 1)� 1����� ;

or equivalently,zf 0(z)

f (z)2 k � P [A;B] : (7.1.4)

It can be easily seen that

f (z) 2 k � UCV [A;B] () zf 0 (z) 2 k � ST [A;B] : (7.1.5)

113

Special Cases

(i) k � ST [1;�1] = k � ST; k � UCV [1;�1] = k � UCV; the well-known classes of

k-uniformly convex and k-starlike functions respectively, introduced by Kanas and

Wisniowska [36, 35].

(ii) k � ST [1� 2�;�1] = SD (k; �) ; k � UCV [1� 2�;�1] = KD (k; �) ; the classes,

introduced by Shams et.al in [101].

(iii) 0 � ST [A;B] = S� [A;B] ; 0 � UCV [A;B] = C [A;B] ; the well-known classes

of Janowski starlike and Janowski convex functions respectively, introduced by

Janowski [30].

Now we discuss some results concerning above classes.

7.2 Main results

Our �rst result is a su¢ cient condition for a function f (z) 2 A to be from k�ST [A;B] :

It gives many known results as special cases as discussed after this result.

Theorem 7.2.1 A function f (z) 2 A and of the form (2:1:1) is in the class k �

ST [A;B] ; if it satis�es the condition

1Xn=2

f2 (k + 1) (n� 1) + jn (B + 1)� (A+ 1)jg janj < jB � Aj ; (7.2.1)

where �1 � B < A � 1 and k � 0:

Proof. Assuming that (7:2:1) holds, then it su¢ ces to show that

k

�����(B � 1)zf 0(z)f(z)

� (A� 1)

(B + 1) zf0(z)f(z)

� (A+ 1)� 1������ Re

"(B � 1) zf

0(z)f(z)

� (A� 1)

(B + 1) zf0(z)f(z)

� (A+ 1)� 1#< 1:

114

We have

k

�����(B � 1)zf 0(z)f(z)

� (A� 1)

(B + 1) zf0(z)f(z)

� (A+ 1)� 1������ Re

"(B � 1) zf

0(z)f(z)

� (A� 1)

(B + 1) zf0(z)f(z)

� (A+ 1)� 1#

� (k + 1)����(B � 1) zf 0 (z)� (A� 1) f (z)(B + 1) zf 0 (z)� (A+ 1) f (z) � 1

����= 2 (k + 1)

���� f (z)� zf 0 (z)

(B + 1) zf 0 (z)� (A+ 1) f (z)

����

= 2 (k + 1)

��������1Pn=2

(1� n) anzn

(B � A) z +1Pn=2

fn (B + 1)� (A+ 1)g anzn

��������

� 2 (k + 1)

1Pn=2

j1� nj janj

jB � Aj �1Pn=2

jn (B + 1)� (A+ 1)j janj:

The last expression is bounded above by 1 if

1Xn=2

f2 (k + 1) (n� 1) + jn (B + 1)� (A+ 1)jg janj < jB � Aj

and this completes the proof.

When A = 1; B = �1 in above theorem, then we have the following known result,

proved by Kanas and Wisniowska in [35].

Corollary 7.2.1 A function f (z) 2 A and of the form (2:1:1) is in the class k � ST; if

115

it satis�es the condition

1Xn=2

fn+ k (n� 1)g janj < 1; k � 0:

When A = 1 � 2�; B = �1 with 0 � � < 1 in Theorem 7.2.1, then we have the

following known result, proved by Shams et.al in [101].

Corollary 7.2.2 A function f (z) 2 A and of the form (2:1:1) is in the class SD (k; �) ;

if it satis�es the condition

1Xn=2

fn (k + 1)� (k + �)g janj < 1� �;

where 0 � � < 1 and k � 0:

When A = 1 � 2�; B = �1 with 0 � � < 1 and k = 0 in Theorem 7.2.1, then we

have the following known result, proved by Selverman in [100].

Corollary 7.2.3 A function f (z) 2 A and of the form (2:1:1) is in the class S� (�) ; if

it satis�es the condition

1Xn=2

(n� �) janj < 1� �; 0 � � < 1:

Theorem 7.2.2 A function f (z) 2 A and of the form (2:1:1) is in the class k �

UCV [A;B] ; if it satis�es the condition

1Xn=2

n f2 (k + 1) (n� 1) + jn (B + 1)� (A+ 1)jg janj < jB � Aj ;

where �1 � B < A � 1 and k � 0:

Proof follows immediately by using Theorem 7.2.1 and (7:1:5) :

Now we �nd the coe¢ cient bound for functions of k�ST [A;B] : It gives many known

results as special cases. Also it gives a re�nement in one already known result.

116

Theorem 7.2.3 Let f (z) 2 k � ST [A;B] and is of the form (2:1:1) : Then, for n � 2;

janj �n�2Yj=0

j�k (A�B)� 2jBj2 (j + 1)

; (7.2.2)

where �k is de�ned by (2:6:3) :

Proof. By de�nition, for f (z) 2 k � ST [A;B] ; we have

zf 0 (z)

f (z)= p (z) ; (7.2.3)

where

p (z) � (A+ 1) pk(z)� (A� 1)(B + 1) pk(z)� (B � 1)

= [(A+ 1) pk(z)� (A� 1)] [(B + 1) pk(z)� (B � 1)]�1

=

�A� 1B � 1 �

A+ 1

B � 1pk(z)�"1 +

1Xn=1

�B + 1

B � 1pk(z)�n#

=A� 1B � 1 �

A+ 1

B � 1pk(z) +A� 1B � 1

1Xn=1

�B + 1

B � 1pk(z)�n

�A+ 1B � 1pk(z)

1Xn=1

�B + 1

B � 1pk(z)�n

=A� 1B � 1 +

�(A� 1) (B + 1)(B � 1)2

� A+ 1

B � 1

�pk(z)

+

(A� 1) (B + 1)2

(B � 1)3� (A+ 1) (B + 1)

(B � 1)2

!(pk(z))

2

+

(A� 1) (B + 1)3

(B � 1)4� (A+ 1) (B + 1)

2

(B � 1)3

!(pk(z))

3 + � � � :

117

If pk(z) = 1 + �kz + � � � ; then we have

p (z) � A� 1B � 1 +

�(A� 1) (B + 1)(B � 1)2

� A+ 1

B � 1

�(1 + �kz + � � � )

+

(A� 1) (B + 1)2

(B � 1)3� (A+ 1) (B + 1)

(B � 1)2

!(1 + �kz + � � � )2

+

(A� 1) (B + 1)3

(B � 1)4� (A+ 1) (B + 1)

2

(B � 1)3

!(1 + �kz + � � � )3 + � � �

=

(A� 1B � 1 +

(A� 1) (B + 1)(B � 1)2

� A+ 1

B � 1 +(A� 1) (B + 1)2

(B � 1)3

�(A+ 1) (B + 1)(B � 1)2

+ :::

�+

�(A� 1) (B + 1)(B � 1)2

� A+ 1

B � 1

+2

(A� 1) (B + 1)2

(B � 1)3� (A+ 1) (B + 1)

(B � 1)2

!

+3

(A� 1) (B + 1)3

(B � 1)4� (A+ 1) (B + 1)

2

(B � 1)3

!)�kz + :::

=

(�2B � 1 �

2 (B + 1)

(B � 1)2� 2 (B + 1)

2

(B � 1)3� :::

)

+

�(A� 1) (B + 1)� (A+ 1) (B � 1)

(B � 1)2

+2 (B + 1)

�(A� 1) (B + 1)� (A+ 1) (B � 1)

(B � 1)3�

+3 (B + 1)2�(A� 1) (B + 1)� (A+ 1) (B � 1)

(B � 1)4��

�kz + :::

=

1Xn=1

�2 (B + 1)n�1

(B � 1)n +

( 1Xn=1

2n (A�B) (B + 1)n�1

(B � 1)n+1

)�kz + � � � :

Now we see that the series1Pn=1

�2(B+1)n�1(B�1)n and

1Pn=1

2n(A�B)(B+1)n�1

(B�1)n+1 are convergent and

converge to 1 and A�B2respectively. Therefore,

p (z) � 1 + 12(A�B) �kz + � � � :

118

Now if p(z) = 1 +1Pn=1

cnzn; then by Lemma 2.10.5, we have

jcnj �1

2(A�B) j�kj ; n � 1: (7.2.4)

Now from (7:2:3) ; we have

zf 0 (z) = f (z) p (z) ;

which implies that

z +1Xn=2

nan zn =

z +

1Xn=2

an zn

! 1 +

1Xn=1

cn zn

!:

Equating coe¢ cients of zn on both sides, we have

(n� 1) an =n�1Xj=1

an�jcj; a1 = 1:

This implies that

janj �1

n� 1

n�1Xj=1

jan�jj jcjj ; a1 = 1:

Using (7:2:4) ; we have

janj �j�kj (A�B)

2 (n� 1)

n�1Xj=1

jajj ; a1 = 1: (7.2.5)

Now we prove that

j�kj (A�B)

2 (n� 1)

n�1Xj=1

jajj �n�2Yj=0

j�k (A�B)� 2jBj2 (j + 1)

: (7.2.6)

For this, we use induction method.

For n = 2;

119

From (7:2:5) ; we have

ja2j �j�kj (A�B)

2:

From (7:2:2) ; we have

ja2j �j�kj (A�B)

2:

For n = 3;

From (7:2:5) ; we have

ja3j �j�kj (A�B)

4(1 + ja2j)

� j�kj (A�B)

4

�1 +

j�kj (A�B)

2

�:

From (7:2:2) ; we have

ja3j �j�kj (A�B)

2

j�k (A�B)� 2Bj4

� j�kj (A�B)

2

j�kj (A�B) + 2 jBj4

� j�kj (A�B)

2

�j�kj (A�B)

2+ 1

�:

Let the hypothesis be true for n = m:

From (7:2:5) ; we have

jamj �j�kj (A�B)

2 (m� 1)

m�1Xj=1

jajj ; a1 = 1:

120

From (7:2:2) ; we have

jamj �m�2Yj=0

j�k (A�B)� 2jBj2 (j + 1)

�m�2Yj=0

j�kj (A�B) + 2j

2 (j + 1):

By induction hypothesis, we have

j�kj (A�B)

2 (m� 1)

m�1Xj=1

jajj �m�2Yj=0

j�kj (A�B) + 2j

2 (j + 1):

Multiplying both sides by j�kj(A�B)+2(m�1)2m

; we have

m�1Yj=0

j�kj (A�B) + 2j

2 (j + 1)� j�kj (A�B)

2 (m� 1)j�kj (A�B) + 2 (m� 1)

2m

m�1Xj=1

jajj

=j�kj (A�B)

2m

(j�kj (A�B)

2 (m� 1)

m�1Xj=1

jajj+m�1Xj=1

jajj)

� j�kj (A�B)

2m

(jamj+

m�1Xj=1

jajj)

=j�kj (A�B)

2m

mXj=1

jajj :

That is,j�kj (A�B)

2m

mXj=1

jajj �m�1Yj=0

j�kj (A�B) + 2j

2 (j + 1);

which shows that the inequality (7:2:6) is true for n = m+ 1: Hence the required result.

121

Corollary 7.2.4 When A = 1; B = �1; then (7:2:2) reduces to

janj �n�2Yj=0

j�k + jjj + 1

; n � 2;

which is the coe¢ cient inequality of the class k�ST; introduced by Kanas and Wisniowska

[35].

Corollary 7.2.5 When A = 1� 2�; B = �1 with 0 � � < 1; then (7:2:2) reduces to

janj �n�2Yj=0

j�k (1� �) + jjj + 1

; n � 2; (7.2.7)

which is the coe¢ cient inequality of the class SD (k; �) ; introduced by Shams et.al [101].

The inequality (7:2:7) gives the better result as compared with that, proved by Owa

et al. [83]. Owa et al. [83] gave the following coe¢ cient bound of SD (k; �) ;

ja2j � 2(1��)j1�kj ;

janj � 2(1��)(n�1)j1�kj

n�2Qj=1

�1 + 2(1��)

jj1�kj

�; n � 3:

9>=>; (7.2.8)

According to (7:2:8) ; for k = 12; � = 1

2; we have ja2j � 2 but from (7:2:7) we have

ja2j � 0:5926 which clearly shows that our result is far better than that of Owa et al.

[83].

When k = 0; then �k = 2 and from Theorem 7.2.3 we get the following known result,

proved in [30].

Corollary 7.2.6 Let f (z) 2 S� [A;B] and is of the form (2:1:1) : Then, for n � 2;

janj �n�2Yj=0

j(A�B)� jBjj + 1

; �1 � B < A � 1:

When A = 1 � 2�; B = �1 with 0 � � < 1 and k = 0 in Theorem 7.2.3, then we

have the following known result, see [24].

122

Corollary 7.2.7 Let f (z) 2 S� (�) and is of the form (2:1:1) : Then, for n � 2;

janj �

nQj=2

(j � 2�)

(n� 1)! ; 0 � � < 1:

Theorem 7.2.4 Let f (z) 2 k�UCV [A;B] and is of the form (2:1:1) : Then, for n � 2;

janj �1

n

n�2Yj=0

j�k (A�B)� 2jBj2 (j + 1)

; (7.2.9)

where �k is de�ned by (2:6:3) :

Proof follows immediately by using Theorem 7.2.3 and (7:1:5) :

Theorem 7.2.5 Let f (z) 2 S: Then f (z) 2 k � UCV [A;B] for jzj < r0 < 1 with

r0 =2�

p3 + �2

1 + �;

where

� =2k + 1� A

2k + 1�B: (7.2.10)

Proof can be done by using similar technique as used in Theorem 4.2.3.

When A = 1; B = �1 and k = 0 in Theorem 7.2.5, then we have the following result,

proved in [24].

Corollary 7.2.8 Let f (z) 2 S: Then f (z) 2 C for jzj < r0 < 1 with r0 = 2�p3:

When A = 1; B = �1 and k = 1 in Theorem 7.2.5, then we have the following result,

proved in [96].

Corollary 7.2.9 Let f (z) 2 S: Then f (z) 2 UCV for jzj < r0 < 1 with

r0 =4�

p13

3:

123

When A = 1 and B = �1 in Theorem 7.2.5, then we have the following result, proved

by Kanas and Wisniowska [36].

Corollary 7.2.10 Let f (z) 2 S: Then f (z) 2 k � UCV for jzj < r0 < 1 with

r0 =2 (k + 1)�

p4k2 + 6k + 3

2k + 1:

Now we have an extension of the Lemma 2.10.1 proved in [33].

Lemma 7.2.1 Let 0 � k <1: Also, let �; 2 C be such that � 6= 0 and Re (�� + ) >

0; where � is de�ned by (7:2:10) : If p (z) is analytic in E; p (0) = 1; p (z) satis�es

p (z) +zp0 (z)

�p (z) + � pk (A;B; z) ; (7.2.11)

where

pk (A;B; z) =(A+ 1) pk(z)� (A� 1)(B + 1) pk(z)� (B � 1)

and if q (z) is analytic solution of

q (z) +zq0 (z)

�q (z) + = pk (A;B; z) ;

then q (z) is univalent, p (z) � q (z) � pk (A;B; z) and q (z) is the best dominant of

(7:2:11) :

Proof follows similarly as given in [33].

Moreover, the solution q (z) is given by

q (z) =

��

Z 1

0

�t�+ �1 exp

Z tz

z

pk (A;B;u)� 1u

du

��dt

��1�

�:

For more details of best dominant, we refer to [33, 53].

124

As a special case, when � = 1 and = 0; we have the function q (z) as

q (z) =

24 1Z0

0@exp tzZz

pk (A;B;u)� 1u

du

1A dt

35�1 : (7.2.12)

Now we see a few applications of the Lemma 7.2.1.

When k > 1; the domain k [A;B] may be characterized by the circular domain

having its diameter end points as the vertices of the oval (or ellipse). As we see that the

vertices of oval (or ellipse) are (�; 0) and (�1; 0) ; where � is de�ned by (7:2:10) and

�1 =2k + A� 12k +B � 1 :

The circle K (X;R) having diameter end points (�; 0), (�1; 0) has its center at

X�4k2�(A�1)(B�1)4k2�(B�1)2 ; 0

�and radius R as

R =2k (A�B)

4k2 � (B � 1)2:

The point z = 1 is contained inside the circleK (X;R) and then the function �k (A;B; z) :

E �! K (X;R) has the form

�k (A;B; z) =2k + (A� 1) z2k + (B � 1) z :

Theorem 7.2.6 Let k 2 (1;1) : Also, let p (z) be analytic in E with p (0) = 1 and p (z)

satis�es (7:2:11) : Then

p (z) �

8>>>>><>>>>>:(1�A)z2k

��1 + (B�1)z

2k

�B�AB�1 �

�1 + (B�1)z

2k

���1; �1 < B < A < 1;

z

(z�k) log(1� zk); A = 1; B = �1;

125

and

Re p (z) >

8>>>><>>>>:A�12k

h�1� B�1

2k

�B�AB�1 �

�1� B�1

2k

�i�1; �1 < B < A < 1;

1

(k+1) log(1+ 1k); A = 1; B = �1:

Proof. Since p satis�es (7:2:11) and for each �xed k; pk (A;B; z) � �k (A;B; z) ; so

p (z) � �k (A;B; z) : This implies from (7:2:12) ;

q (z) =

24 1Z0

0@exp tzZz

�k (A;B;u)� 1u

du

1A dt

35�1

=

24 1Z0

0@exp tzZz

2k+(A�1)u2k+(B�1)u � 1

udu

1A dt

35�1

=

24 1Z0

0@exp tzZz

A�B

2k + (B � 1)udu

1A dt

35�1

=

24 1Z0

�exp

��A�B

B � 1

�log

2k + (B � 1) tz2k + (B � 1) z

��dt

35�1

=

24 1Z0

�2k + (B � 1) tz2k + (B � 1) z

�A�BB�1

dt

35�1

=

24� 1

2k + (B � 1) z

�A�BB�1

1Z0

(2k + (B � 1) tz)A�BB�1 dt

35�1

=(A� 1) z

(2k + (B � 1) z)B�AB�1

�(2k + (B � 1) z)

A�1B�1 � (2k)

A�1B�1��1

=(1� A) z

2k

(2k + (B � 1) z)A�BB�1

(2k)A�BB�1

241� �1 + (B � 1) z2k

�A�1B�135�1

126

=(1� A) z

2k

�1 +

(B � 1) z2k

�A�BB�1

241� �1 + (B � 1) z2k

�A�BB�1 +1

35�1

=(1� A) z

2k

"�1 +

(B � 1) z2k

�B�AB�1

��1 +

(B � 1) z2k

�#�1:

Also we have

Re p (z) > q (�1) =

8>>>><>>>>:A�12k

h�1� B�1

2k

�B�AB�1 �

�1� B�1

2k

�i�1; �1 < B < A < 1;

1

(k+1) log(1+ 1k); A = 1; B = �1;

for z 2 E:

The above Lemma 7.2.1 and Theorem 7.2.6 have been proved only for A = 1; B = �1

by Kanas [33].

Theorem 7.2.7 If f (z) 2 A satis�es the inequality

Re

( zf 00(z)f 0(z)

zf 0(z)f(z)

� 1

)<3� �

2� �;

where � is de�ned by (7:2:10) ; then f (z) 2 k � ST [A;B] ; k � 0; with �1 � B < 0 and

B < A � 1:

Proof can be done by using similar technique as used in Theorem 4.2.1.

From the Theorem 7.2.7, we see that when A = 1; B = �1 and k = 1; we have the

following result which is the special case (when p = 1) of the result proved by Al-Kharsani

et.al [6].

Corollary 7.2.11 If f (z) 2 A satis�es the inequality

Re

( zf 00(z)f 0(z)

zf 0(z)f(z)

� 1

)<5

3;

127

then f (z) is uniformly starlike in E (that is f (z) 2 1� ST ).

7.3 Conclusion

We introduced new geometrical structures of oval and petal type shaped as image domain

and de�ned the classes of functions which give these types of mappings. We introduced

the domain k [A;B] which represents the oval and petal type regions. Di¤erent graphical

views of this new domain for speci�c values of parameters are shown in order to have

better understanding of the behaviour of this domain k [A;B] : Moreover, two new

classes k�UCV [A;B] of k-uniformly Janowski convex and k�ST [A;B] of k- Janowski

starlike functions are de�ned and a number of already known classes of analytic functions

are obtained from these classes as special cases. We investigated certain properties of

these classes. We improved the coe¢ cient bound of the class SD (k; �) which was proved

by Owa et al. [83].

128

Chapter 8

Conclusion

129

This research is carried out on analytic functions associated with conic domains. In this

thesis, we generalized the conic domains and circular domains. We removed the de�ciency

in already known conic domains and made it able to magnify as well as contract. We also

introduced new petal and oval type domains by combining conic and circular domains.

We de�ned certain classes of analytic functions which give these types of mappings. Many

of our results are re�nement of already known results. The concept of our new introduced

domains and their respective classes of analytic functions will open a new direction of

future research. Various di¤erential and integral operators can be applied to introduce

and study new classes of analytic functions and many other geometrical properties can

also be studied.

It is worthy to mention that most of this work has been published in well reputed

journals as listed below.

1. Khalida Inayat Noor and Sarfraz Nawaz Malik (2012). On generalized bounded

Mocanu variation associated with conic domain, Math. Comput. Modell. 55,

844-852.

2. Khalida Inayat Noor and Sarfraz Nawaz Malik (2011). On a new class of analytic

functions associated with conic domain, Comput.Math. Appl. 62, 367�375.

3. Khalida Inayat Noor, Sarfraz Nawaz Malik, Muhammad Arif and Mohsan Raza

(2011). On Bounded Boundary and Bounded Radius Rotation Related with Janowski

Function, World Appl. Sci. J. 12 (6), 895-902.

4. Khalida Inayat Noor and Sarfraz NawazMalik (2011). On coe¢ cient inequalities

of functions associated with conic domains, Comput.Math. Appl. 62, 2209�2217.

Although, writing up this thesis was not an easy task, but the scholarly, perceptive

and insightful guidance of Prof. Dr. Khalida Inayat Noor made it possible for me. This

work is rich with new ideas and can motivate many researchers working in this �eld.

130

Chapter 9

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