Chapter 1

INTRODUCTION, DEFINITIONS AND

FUNDAMENTAL RESULTS

Section 1

1.1 Preliminaries and introduction

The main aim of this chapter is to give the list of all the definitions, lemmas, Theorems and basic

results which we require during the course of our research work. The relevant references are also

mentioned at the end of the chapter to give more details, for example, Duren [17] and Miller and

Mocanu [121].

GFT is a sub- branch of complex analysis. The Study of GFT involves finding the relationship

between the analytical properties of the function of Complex variable ( ), and the

geometrical properties of the image domain = ( ) where D is the unit disc. We attempt to

investigate some geometrical aspects of Univalent (or schlicht i. e. a single valued function) and

multivalent functions. We will obtain some nice results of coefficients estimates, convolution

property (an essential quality), growth and distortion bounds, extreme points, criterion for

univalency, p-valency and extremal functions for these classes. The study will continue the

investigation of several properties of Holomorphic (an analytic) and Univalent (or schlicht i. e. a

single valued function) functions and the different sub classes like, star like, convex, close-to-

convex, uniformly star like and convex, parabolic star like and convex etc. Application of

fractional derivate Operator, fractional Integral Operator, and hyper geometric functions to

different sub classes of polylogarithms and univalent (or schlicht i. e. a single valued function)

and multivalent functions is also of interest to us. The study of differential sub- ordinates and

criteria for univalency to various Sub classes of univalent (or schlicht i. e. a single valued

function) and p-valent functions is also under consideration. Using GFT and utilizing the theory

of Complex Analysis we will try to find the connections with other branches of mathematics and

applications in science and engineering. In this venture we will define several sub classes of

univalent (or schlicht i. e. a single valued function) and multivalent functions with multiple

properties.

Geometric functions theory is quite old and has become one of the most outstanding branches of

Complex Analysis. This study involves finding the relationship between the analytical properties

and geometrical properties D = u (D) where D is the unit disc. Attempts to solve the numerous

conjectures in geometric functions theory have resulted in enriching the classical Geometric

Function Theory in several directions. Geometric Function Theory primarily comprises of

univalent (or schlicht i. e. a single valued function) and multivalent functions and its role in the

development of Complex Analysis. Our aim is to investigate some geometrical aspects of

univalent (or schlicht i. e. a single valued function) and multivalent functions.

The theory of conformal mappings is intimately connected with the theory of boundary value

problems for harmonic functions. Hence GFT has many applications in mathematical physics.

The need also arises for good numerical methods for construction of conformal mappings.

However, this interplay works only in two-dimensions. In three-dimensions due to the classical

theory of Liouville, there are only few and trivial conformal mappings. In higher dimensions the

powerful tool of conformal mapping fails. GFT has and continues to have a profound impact on

other branches of Mathematics like:

1. Complex Analysis

2. Harmonic Analysis

3. Functional Analysis

4. Algebra

5. Partial Differential equations

6. Dynamics

7. Geometry

8. Topology

9. Global Analysis

There are many conjectures in Mathematics which have been solved by the use of Geometric

functions theory, for instance the, “Biberbach Conjecture”. This conjecture has been solved for

some values of n and for all values of n for certain sub classes of univalent (or schlicht i. e. a

single valued function) functions, but the full conjecture still remains open.

The first stirrings of function theory were found in the 18th century with L. Euler. Modern

function theory was developed in 19th century. The pioneers of the subject were A.L Cauchy, B.

Riemann and K. Weierstrass. In 1851, B. Riemann showed that there always exists a

Holomorphic (an analytic) function f that maps any given simply connected domain D¹.in the z -

plane on to a given simply connected domain D2 in the w- plane. This original version of

Riemann mapping theorem gave rise to the birth of Geometric Function Theory. The theory of

univalent (or schlicht i. e. a single valued function) functions was initiated by P. Koebe in 1907.

Most of this field is concerned with S that is class of Holomorphic (an analytic) and Univalent

(or schlicht i. e. a single valued function) functions in E = z: | z |<1, normalized in standard

way. One of the major problems of the field had been the “Biberbach Conjecture” dating from

the year 1916, which asserts the modulus of Taylor series coefficients of each function in class S.

For many years it is proved in 1985. The main idea of the field is closely related to conformal

mappings in the open unit disc or half plane, which have many applications in Engineering,

Physics, Electronics, Medicines and other branches of applied mathematics. In the course of

tackling “Biberbach Conjecture”, new classes of univalent (or schlicht i. e. a single valued

function) and multivalent Holomorphic (an analytic) functions such as classes of convex and star

like functions etc. were defined and some nice properties of these classes were widely

investigated. In 1952, W. Kaplan [51] introduced the most famous class K of close-to-convex

univalent (or schlicht i. e. a single valued function) functions and gave its geometrical

interpretation. Well- known two Sub classes of the class S, namely the class of star like functions

and the class of convex functions are related to each other by Alexander relation. Later Libera

introduce an Integral Operator and showed that these two classes are closed under this operator.

Bernardi gave a generalized operator and studied its properties. Recently Ruscheweyh, Noor and

many others defined new operators and studied new classes of Holomorphic (an analytic) and

univalent (or schlicht i. e. a single valued function) functions which generalizes a number of

previously known classes of Holomorphic (an analytic) and univalent (or schlicht i. e. a single

valued function) functions.

Books by Schaeffer and Spencer, and Jenkins cover in great detail special areas in the field. The

texts by Hayman and Goluzin give a comprehensive survey, and Chapter 6 of Research problems

in function theory contains enough open problems to occupy our efforts for some time. Further

guidance in the field is supplied by the survey articles of Bernardi, Krzyz and Hayman.

The specialist must be grateful to S. D. Bernardi, who has devoted much of his time to preparing

an exhaustive bibliography of the subject. The first volume covers the period from 1907 to 1965

and lists 1694 papers covering the publications of at least 570 authors. The second volume

covers the period from 1966 to 1975 and lists 1563 papers, each of these, 2 volumes has an

extensive index which lists subtopics in this field and those papers that touch on each subtopic.

Thus it is an easy matter for the specialist to determine the status of any problem up to the year

1975. In addition, these two volumes give references to the many survey articles on the subject

that have appeared during this period. Here we cite only three such; one by D. Campbell, a

second by Anderson, Barth and Brannan I, and third, the article by the author. It is the purpose of

this paper to review this third survey and bring it up to date by reporting on the progress made on

the problems mentioned there.

Definition 1.1.1: A set CCE , denoting the set containing complex numbers. We can say is

star like with respect to ∈ if the linear segment joining to each point ∈ entirely in .

In more picturesque language, the requirement is that every point of be visible from . The set

E is said to be convex if it is star like with respect to each of its points.

Definition 1.1.2 A function = ( ) is called as Holomorphic (an analytic) in a domain if it

is differentiable at every point in that domain .

Definition 1.1.3 A Holomorphic (an analytic) function is called as univalent (or schlicht) in

domain if it does not take the same value twice that is )()( 21 zuzu for all pairs of distinct

points 1z and 2z in D1. In other words, u is one-to-one (or injective) mapping of domain D

onto another domain D2, if )(zu assumes the same value more than one, i. e. here after it is to be

taken as )(zu is said to be multivalent (p-valently) in the unit disk D with the form

2

u(z)n

nn zaz ∈ .(0 ≤∝< 1; ∈ ) (1.1.1)

Definition 1.1.4: A Holomorphic (an analytic) function Au is called as star like of order α iff′( )( ) >∝ (1.1.2)

Definition 1.1.5: A Holomorphic (an analytic) function Au is called as convex of order α if &

only if 1 + ′′( )′( ) >∝ ).;10( Uz (1.1.3)

That is CCSS )0(,)0( and ASC and the Koebe function is star like but not

convex, where the Koebe function is given by( ) = (1 − ) = ∞

This is the most famous function in the class A which maps U onto C minus a slit along the

negative real axis from4

1 to .

Definition 1.1.6: A function u Holomorphic (an analytic) in the unit disk U is known as close-

to- convex function of order )10( if ∃ a convex function )(zv s. t.

,)('

(z)u'Re

zv

∀ Uz . (1.1.4) Where )(K is the class of close-

to- convex functions of order ∝, normalized by usual

Condition .01)0(')0( uu In terms of argument, we can write the above condition as

,2)('

)('arg

zv

zu ∀ Uz . (1.1.5)

These functions are connected by the relation .KSC

Definition 1.1.7: A function ( ) is called close- to- convex of order if & only if ∃ a

function Sg satisfying

,)(

)('Re

zv

zuze i ,Uz ,10

22

.

Definition 1.1.8 If ( ) = + ∑∞ ∈ , = 1,2,3, … (1.1.6)

I. e. here after it is to be taken as we say that the function u is p-valently star like of order and

p-valently convex of order ∝ where (0 ≤∝< ) respectively if & only if′( )( ) >∝ & 1 + ′′( )

′( ) >∝.

Definition 1.1.9: Letting ∑ as class of Holomorphic (a regular) functn in the punctured disk

given by : 0 1 .U z z

I. e. here after it is to be taken as

qn

nn

q zazu(z) (1.1.7)

We say that Holomorphic (an analytic) function u is p -valently Meromorphic (analytic except

for isolated singularities i. e. poles) star like of order )0( p if & only if

u(z)

)('Re

zuz ).( Uz

Also u is p-valently Meromorphic (analytic except for isolated singularities i. e. poles) convex

of order ∝ where (0 ≤∝< ) if & only if

,)('

)(''1Re

zu

zuz ).( Uz

Definition 1.1.10: The Convolution or Hadamard Product (or convolution) for functions u and

v denoted by vu is defined as follows for the functions in

)( pA and qrespectively.

(1) If

1

,u(z)qn

nn

q zaz

1

,)(qn

nn

q zbzzv

I. e. here after it is to be taken as

.))(( nnn

q zbazzvu

(2) If ( ) = + ∑∞ ( ) = + ∑∞

I. e. here after it is to be taken as

qn

nnn

q zbazzvu .))((

Definition 1.1.11 The Quasi- Hadamard product (or Convolution) of the function )(zu j for

nj ,.....,3,2,1 Satisfying( ) = + ∑ ,∞ ),........,3,2,1,,;0( , njNnqa kj (1.1.8)

1

,,2,121 ........)(.........)()(qk

kknkk

qn zaaazzuzuzu (1.1.9)

The Quasi-Hadamard product (or Convolution) was investigated and studied by Owa [107],

Schild and Silverman [66] and Srivastava and Aouf [61]. By using the definition of Convolution

or Hadamard product (or Convolution) of two functions we define the familiar Ruscheweyh

derivative [113]. Let be the class representing the functions given byℎ( ) = 1 + ℎ + ℎ +∙∙∙Let be Holomorphic (an analytic) in a domain D and [ℎ( )] > 0.Represented by : → .I. e. here after it is to be taken as( ) = ( ) ∗ ( ) (−1 < ). (1.1.10)

The symbolic notation is known as the Ruscheweyh derivative operator of function of

order.

Hence = ,= ′

& ∝ ( ) = ( ∝ ( ))(∝)∝ (∀ ∝ = ∪ 0). (1.1.11)

Definition 1.1.12 Let ( ) and ( ) be Holomorphic (an analytic) in the unit disk , i. e. here

after it is to be taken as ( ) is said to be sub ordinate to ( ) denoted by ( ) ≺ ( ) if ∃Schwarz functn ),(zw i. e. Holomorphic (a regular) in the unit disk , where at = 0 is 0 &

mod of ( ) less than 1. ( ) = [ ( )].Hence it is given as

)()( zvzu ∈ )0()0( vu , and ( ) contained in ( ).If )(zv is univalent (or schlicht i. e. a single valued function) in D

i. e. here after it is to be taken as

)()( zvzu , ∈ )0()0( vu and ( ) contained in ( ).Definition 1.1.13 Let us assume the Topological vector space over the vector field C and let it

be E a subset of X. A point Ex is called an extreme point of E if it has no representation of

the form ,10,)1( lzttyx as a proper convex combination of two distinct points y and

z in .E

Definition 1.1.14 Let EX , be as in Definition 1.1.13, i. e. here after it is to be taken as the

convex hull of E is the smallest convex set containing E and the closed convex hull of E is the

smallest closed convex set consisting ;E which is the closer convex hull of ,E we denote the

closed convex hull of E by .

Definition 1.1.15 Radius of star likeness of a function f is the largest , where0 < < 1the unit disk , for which it is star like in .0rz

Definition

1.1.16 Radius of convexity of a function f is the largest , where 0 < < 1, for which it is

convex in .1rz

Definition 1.1.17 The weighted mean )(zh of functions ( ) & ( ), defined by

)()1()()1(2

1)( zvjzujzh .

Also

q

jj zu

qzh

1

)(1

)(

ℎ( ) Is the arithmetic mean of )(zu j ( 1, 2,......, ).j q

Definition 1.1.18 Au is known as uniformly convex denoted by UCV if & only if ∀ circular

arc contained in with center also in , the image arc )(f is convex. It is well- known that

u UCV if & only if 1 + ′′( )′( ) > ′′( )

′( ) ∈ .Definition 1.1.19 The function Au is said to be -uniformly convex in U denoted by

)0( UCV if & only if1 + ′′( )′( ) >∝ ′′( )

′( ) ∈ .Geometrically we can say that − is a collection of functions of which map each circular

arc with center of the point ),( C onto an arc which is convex arc, (1 ),UCV UCV

this class is introduced by [92].

Definition 1.1.20 The function Au is said to be -star like ( 0 ) function in U denoted by

ST if & only if 1 + ′′( )′( ) >∝ ′′( )

′( ) − 1 ∈ .The class ST is a naturally way emerged as the class of functions with the property (an

essential quality) that UCVv if & only if .)(' STzzv this class is introduced by

Kanas and Wisniowska [92].

Definition 1.1.21 Let (0 < < 1) and

z

z dttz

tu

dz

dzfD

0 )(

)(

)1(

1)(

(1.1.12)

Where called as fractional derivative of order. Function u(z) is as in defi-nition1.1.20 & the

multiplicityof ( − ) λ is eliminated like Definition1.1.21.

Definition 1.1.22: If ivuf be harmonic, i. e. here after it is to be taken as

We can find Holomorphic (an analytic) function HG, s. t. Gu Re ,ImHv ∴ ,22

HGHGghf

Here ℎ is Holomorphic (an analytic) part of , that is co-Holomorphic (an analytic) part of f .

Definition 1.1.23 The harmonic = ℎ + is sense- preserving and locally injective iff

,0)()()(22 zgzhzf where f denotes the Jacobin of )( zf . If

ghf Is harmonic, sense-preserving and injective, i. e. here after it is to be taken as we can

say that f is harmonic univalent (or schlicht i. e. a single valued function) function.

Section 2

1.2 Basic Lemmas and Theorems

Theorem 1.2.1: If 0 and 10 and ,R i. e. here after it is to be taken as

1Re ww if & only if 1+∝ −∝ > ,Where w be any Complex number.

Theorem 1.2.2: With the same condition as in above theorem 1.2.1, w if & only if

)1()1( ww .

Next result is by using Alexander’s theorem.

Theorem 1.2.3 If be a Holomorphic (an analytic) function in U with 01)0(')0( uu , i. e.

here after it is to be taken as Cu if & only if .' Szu

Theorem 1.2.4 (Distortion Theorem) For each Au(1 − )(1 + ) ≤ | ( )| ≤ (1 + )(1 − ) , | | = < 1,∀ .z U Where ≠ 0 and equality occurs if & only if suitable rotation of the Koebe function is

function . We say upper and lower bounds for )(' zu as Distortion bounds.

Theorem 1.2.5 (Growth Theorem) For each Au

( ) )(zu ( ) , rz , < 1.∀ .z U Where ≠ 0 and equality occurs if & only if suitable rotation of the Koebe function is

function .

Lemma 1.2.1 (Schwarz Lemma): Let u be Holomorphic (a regular) in U with (0) = 0 and( ) < 1 in i. e. here after it is to be taken as ′(0) ≤ 1 and | ( )| ≤ | | in D strict

inequality holds in both estimates unless u is rotation the disk .)( zezu i

Lemma 1.2.2 Let us consider the function ( ) which is Holomorphic (an analytic) in as well

as convex in D with (0) = 1. Also assume∅( ) = 1 + + +⋯ (1.2.1)

Where function ∅( ) is Holomorphic (an analytic) in D if

)()('

)( zhz

zz

0;0)Re( , (1.2.2)

z

kk zhdtthtzk

zz0

111 )()(

1)()(

),( Uz (1.2.3)

Which shows that ∅( ) is the dominant of (1.2.1).

Lemma 1.2.3 ∀ real or imaginary numbers , and c )( 0Zc ,

1

1211 );;,(

)(

)()()1()1(

o

abcb zcbaFc

bcbdtzttt ;0)Re()Re( bc (1.2.4)

)1

,;,()1();;,( 1212

z

zcbcaFzzcbaF a ; (1.2.5)

);;1;,1();;1,();;,( 121212 zcbaFc

azzcbaFzcbaF

(1.2.6)

)2

1()

2

1(

)2

1(

)2

1;

2;,(12

ba

bacba

baF

(1.2.7)

Lemma1.2.4 Let )(z be Holomorphic (an analytic) in D with

1)0( And 1Re ( ) ,

2z ∀ ∈ .

I. e. here after it is to be taken as for any function )(zY Holomorphic (an analytic) in D,

))(( DY which is contained in convex hull of ).(DY

Lemma 1.2.5 Let ghf in H be of the form

,)(2

n

nn zazzh ,)(

2

n

nn zbzg 11 b (1.2.8) and if

1)()(1n

nn bnan ,

I. e. here after it is to be taken as f is harmonic, orientation preserving univalent (or schlicht i. e.

a single valued function) in U.

Lemma 1.2.6 Let us assume ghf belonging to H be of the form (1.1.22) and if

2111

nnn b

na

n

110 (1.2.9)

I. e. here after it is to be taken as f is harmonic, sense preserving, Univalent (or schlicht i. e. a

single valued function) and star like of order in .U

1.3 Scope of the research work

The work in this thesis basically involves the study of some sub classes of univalent (or schlicht)

Holomorphic (an analytic) functions; Univalent (or schlicht) and Multivalent Meromorphic

(analytic except for isolated singularities i. e. poles) functions using various fractional calculus

operators and subordination principle.

In Chapter 2, we have investigated two Sub classes )(* pU and * ( ),p which are sub classes

of Meromorphic (analytic except for isolated singularities i. e. poles) univalent (or schlicht i. e. a

single valued function) in unit = : 0 < | | < 1 and some well-known properties of above

two sub classes are studied. Let which represents ( ) given as

1

1)(

n

nn za

zzu

These functions are univalent (or schlicht i. e. a single valued function) as well as Holomorphic

(an analytic) in D given as above.

u Contained in ( ). Here ( ) is the class of Mero morphic (analytic except for

isolated singularities i. e. poles) univalent (or schlicht i. e. a single valued function) star like

functions in the disk D if & only if

)(

)('Re

zu

zzu

.10; Dz

A function u is contained in ( ). where ( ) is the class of Meromorphic (analytic

except for isolated singularities i. e. poles) univalent (or schlicht) convex functions in the

punctured disk in D if & only if

)(

)(1Re

,

,,

zu

zzu

.10; Dz

The classes ( ) and ( )U and the functions in are introduced and discussed by the

researchers [5], [32], [50], Mogra [62], Uralegaddi and Somanatha [67], Srivastava and Owa

[59], Yang [71] and Khairnar [26]. Now we discussed about some sufficient conditions for )( zf

which are in )(U and )( obtained by using Coefficient inequalities we introduced two

more new Sub classes ∑ ( , , ) and of ∑ ( , , ) Meromorphic (analytic except

for isolated singularities i. e. poles) univalent (or schlicht i. e. a single valued function) functions

which are defined by using differential Operator. Using the differential subordination principle

we studied various inclusion relationships and its properties. Those properties are defined by

using differential operator. Some of the outcomes which are concerned with Subordination,

Convolution properties and coefficient estimates; Distortion Theorem, integral representation etc.

are studied. Authors have worked on familiar phenomena of Holomorphic (an analytic) functions

to the sub classes of Meromorphic (analytic except for isolated singularities i. e. poles) univalent

(or schlicht i. e. a single valued function) functions. That phenomenon is of (n, ) −neighborhoods. Let be Holomorphic (an analytic) function in the unit disk defined below= ℎ ℎ ℎ 1.∴ = ∈ ∶ = 0 0 ( ) < 1, ∀ The class of Schwarz function is s. t. ∀ ( 0 ≤ < 1).Let ( ) = ∶ (0) = 1and ( ) > , .

It is simple to prove i.e. ∈ ( ) ⇒ ( ) ∈ .A function ∈ ∑ is Meromorphic (analytic except for isolated singularities i. e. poles)

Univalent (or schlicht i. e. a single valued function) star like of order ∝ (0 ≤∝< 1). If( )( ) < − ∀ ,The classes of all such functions are given by ∑ (∝). If ∈ ∑ is given by above equation &contained in ∑ is given by( ) = + ∑ .I. e. here after it is to be taken as the Convolution of functions and is defined by

( ∗ )( ) = + ∑ a ∀ Contained in ∗.∀ ∈ , the differential operator as given belowv(z) = ( ) ( ) = ( ) = ( − )( − )[ ( )]+ ( ) ( ) ( ) + ( ) ( ) ( ).∴ ( ) = ( )Where it is obvious,0 < ≤ , 0 ≤ ξ < 1, ≥ 1, 0 ≤ < 1, 0 ≤ < 10 ≤∝< 1, 0 < ≤ 1 ∈ .If the function ∈ is given by above Equations i. e. here after it is to be taken as( ) = + ∑ ( , , , , , 1)( & contained in D∗),Where it is obvious, ( , , , , , 1)= 1 + ( + ) ( ) ( )

+( + + 1)( − ⟨ + ς⟩ + ς)] .From above equation it implies that ( ) as Hadamard product (or Convolution) is given as

follows ( ) = ( ∗ ℎ)( )ℎ( ) = + ∑ ( , , , , , 1) .Note that, the case = and = of was investigated by H. M. Srivastava and Patel

[60]. For Differential operator ( ).Definition 2.7.1 ∈ ∑ is in the class ∑ ( , , ) if & only if

( )( ) ( )( )( ) ( ) ( ) < , ∀z ∈ U∗.

Definition 2.10.1 For = ( ) ( )( )( ) ( )( ) > 0 And non negative sequenceS = s .

Where it is obvious that,= [ ( ) | | ( )] ( , , , , , )( ) [ ] (0 ≤ < 1, 0 < ≤ 1).( ) Of a function ∈ ∑ of the form as defined above is( ) = ∈ ∑ : ( ) = + ∑∑ | − | ≤ .For = N, def 2.7.4 corresponds to the ( , ) −neighborhoods derived by [56]. Using the

Definition 2.10.1, we are going to prove some properties and results for ( , ) − neighborhoods

of ∑ (∝, , ).

In Chapter 3, we have derived two sub classes )(* qU and *( ),q these functions are

Meromorphic (analytic except for isolated singularities i. e. poles) multivalent functions in ∗where * : 0 1D z z is punctured unit disk. The properties like Distortion Theorems,

Convolution of functions, etc. which are belonging to these classes are obtained. Let p

be the

class of ( ) given by

11

1

1( ) ,q n

q nqn

u z a zz

∀ ∈ .Functions given by above defined class are Holomorphic (an analytic) and multivalent in the disk

defined as 10: zzD . A function q

q being in )(qU of order in D if &

only if

)(

)('Re

zu

zzu,

.;,0; NqqDz

Moreover, q

u is called as class of Meromorphic (analytic except for isolated singularities i.

e. poles) Multivalent in )(q of order ∝ in D iff

)('

)(''1Re

zu

zzu .;,0; NqqDz

The classes )(),( qqU and another functions like q

have been studied and derived briefly

by researchers [60], [61], [18], [5], [19], [71], [7], [62], [50], [59], [32], [67], Yang [71] [26] and

[27]. In this chapter we are deriving some required necessary and sufficient conditions for )(zu

to be in )(qU and ).(q So that we get by making use of the coefficient inequalities. In this

chapter we introduced some more new Sub classes ∗( , ), , ( , , , , , ; ),( , , , , , ; ), and ℎ ( , , , , , ; ), as well as , ( , , , , , ; ),( , , , , , ; ), and ( , , , , , ; ) of Meromorphic (analytic except for isolated

singularities i. e. poles) multivalent functions in ∗ = : 0 < | | < 1 = \0. By applying

Differential sub-ordinates’ method, we obtained some certain properties of Meromorphic

(analytic except for isolated singularities i. e. poles) Multivalent functions like Inclusion

relationships Integral Representation and Convolution properties using Differential

Subordination. Let p

be the class of ( ) given by

1

11

1)(

n

nqnpq

zaz

zu . ∀ ∈ .These functions are Holomorphic (an analytic) & Multivalent in ∗ where ∗ ℎ of all z

s. t. 0 < | | < 1 excluding 0. Let and be two functions which are Holomorphic (an analytic)

in , i. e. here after it is to be taken as we have observed ( ) ≺ ( ) i.e. Here ( ) is sub

ordinate to ( ) in if and only if ∃ a Holomorphic (an analytic) function ℎ( ) in , s. t.|ℎ( )| ≤ | | and ( ) = [ℎ( ) ] ∀ ∈ . If ( ) is univalent (or schlicht i. e. a single valued

function) in i. e. here after it is to be taken as( ) ≺ ( ) ∀ ∈ ⟺ (0) = (0) & ( ) ⊂ ( ).

Let us consider ( ) Holomorphic (an analytic) in D∴ ( ) = 1 + + +⋯⋯⇒ ( ) ≺ ∀ ∈ .( ) − < ( ), − ≤ < ≤ .Re ( ) > , (2 = −1, ∈ ).

Recently, several authors proved some interesting properties of Meromorphic (analytic except

for isolated singularities i. e. poles) multivalent functions. In this chapter, we have proved few of

the subordination properties for the class S.When, ( ) = ( ) = + ∑ , ∀ ∈ .The convolution of functions ( )& ( ) is as given below( ∗ )( ) = ( ) = + ∑ ,∀ ∈ , ∈ & ∈ .We define a linear operator by ( , , , , , ) ( ) as given below( , , , , , ) ( ) = + ∑ ( )( )( ) + 1= , , , , ,, ∗ ( ).

Where,

, , , , ,, ( ) = + ∑ ( )( )( ) + 1 .Throughout this paper for our convenience we are taking( )( ) = ( , , , , , ) ( ) = ( )∈ , ∈ = ∪ 0, ∈ , ≥ 0, ≥ 0, 0 < ≤ .Simply we can say ( )( ) ( , , , , , ) ( )= ( , , , , , ) ( ) − [ ( )( ) + 1] ( , , , , , ) ( )∴ ( ) = ( ) − ( + 1) ( )We note that ( , , , , , ) ( ) = ( )

& , 1,1,1, , 1 ( ) = ( ) = ( + 1) ( ) + ( ).The above Operators which are Holomorphic (an analytic) in gives ( ) > 0, ∀ . ∈ , = (2 ),

, [ , , , , , ; ] = ∑ = +∙∙∙ ( ∈ )., [ , , , , , ; ] = ( ) + ( ) = +∙∙∙ ( ∈ ).

ℎ , [ , , , , , ; ] = ( ) + (− ) = +∙∙∙ (ℎ ∈ ).For = 1 we have, , [ , , , , , ; ] = ( ).In Chapter 4, we have obtained two new Sub classes ∑ ( , , ) and ∑ ( , , ),using the Differential Subordination principle; we derived various sub classes derived using

differential operator. Here in this chapter few of the outcomes concerned with subordination

properties like Distortion Theorem, coefficient estimates, integral representation, convolution

properties etc. are derived and studied. We extended our discussion regarding well- known

theme of ( , ) − neighborhoods of Holomorphic (an analytic) functions. Let à be the class of

Holomorphic (a regular) in = ∈ : | | < 1. If= ∈ ∶ = 0 0 ℎ 1.Meromorphic (analytic except for isolated singularities i. e. poles) star like function ∈ ∑ if− ( )( ) >∝ ∀ ∈ .above defined class of functions represented by ∑ (∝). If ∈ ∑ i. e. here after it is to be taken

as the above inequality & ∈ ∑ is denoted by( ) = + ∑The convolution of functions given as follows( ∗ )( ) = + ∑ ≥ 0∀ ∈ & ∈ ∗, ∀ ∈ ,

The differential operator is defined as given below( ) = ( )( ) = ( ) = ( − )( − ) ( )+ ( ) ( ) [ . ( )] + ( ) ( ) ( ).( ) = ( ) ,

Where it is obviously for all,0 < ϱ ≤ , 0 ≤ ξ < 1, τ ≥ 1, 0 ≤ ς < 1, 0 ≤ η < 10 ≤∝< 1, 0 < β ≤ 1, ∈ N .

Function ∈ is given by above equations i. e. here after it is to be taken as we obtain( ) = + ∑ ( , , , , , )∀ , ∈ & ∈ ∗.Where it is obviously for,

( , , , , , ) = 1 + ( + ) ( ) ( )+( + + 1)( − )( − ) . It

follows that ( )( ) = ( ∗ ℎ)( )∴ ℎ( ) = + ∑ ( , , , , , ) .The case = and = of the diff. operator , was investigated by researchers [60]. By

using the diff. operator ( ) for = 1was considered in [60]. For differential

operator ( ),

Definition 4.1.1 If function ∈ ∑ being in ∑ ( , , ) iff the inequality condition given

below is satisfied

( )( ) ( )( )( ) ( ) ( ) < , ∀ z ∈ U∗.

Where, 0 < ≤ , 0 ≤ < 1, ≥ 1, 0 ≤ ≤ 1,0 ≤ < 1, 0 ≤∝< 1, 0 < ≤ 1 and ∈ .Note that a class ∑ (∝, , ) for = 1 = 0 is the class of star like Meromorphic

(analytic except for isolated singularities i. e. poles) functions of order ∝ (0 <∝< 1)which are investigated by [46]. It is simple to verify for = 0 = 1, the class∑ (∝, , )converts to the class ∑ (∝).∗ Let us assume other subclass ∑ defined by

∑ (∝, , ) = ∑ (∝, ) ∩ ∑ (∝, , ). T

The main purpose of this chapter is to represent and study the classes represented by

∑ (∝, , ) & ∑ (∝, , ).∗ here in this topic we have discussed and derived some well-

known properties of the class ∑ (∝, , )and properties of the class ∑ (∝, , ), as

well as Neighborhoods and partial sums.

In Chapter 5, we have defined a subclass of multivalent function of the form given as

2

11)(

n

nn zazzu , .01 na .

And some results of Convolutions with Negative coefficients. We have also obtained the

Coefficient bounds and discuss for Sharpness of the results obtained

5.1 Sub classes of Multivalent Functions with Negative Coefficients

Let S be class of multivalent normalized functions as given below

.)(2

n

qnqn

q zazzu

Holomorphic (an analytic) in D= 1; zz .Let T be the subclass of S as follows

2

)(n

qnqn

q zazzu .0qna

Let K= w; Holomorphic (an analytic) in E, s. t. w at z=0 is 0 and mod of w (z) is less than 1 in

E Also let ( , ) gives the class of Holomorphic in which are as follows

)(1

)(1

zBw

zAw

.11 BA

Here w (z) contained in K. Now we are going to study another subclass of T and we denote it by

1T , which consists of functions of the form as given follows

2

)(n

qnqn

q zazzu .0qna

We define,

.),(&;),('

*

BAGf

zfSffBAS

And, .),()(

&;),('

''

BAGf

zfSffBAH further

we will define

),(&;),('

1*

1 BAGf

fzHffBAT

And .),()'(

&;),('

'

11

BAGf

zfHffBAC

If2

( ) &q n qn q

n

u z z a z

.)(2

n

qnqn

q zbzzv

With .0&0 qnqn ba

Their Convolution is defined by )(*)()( zvzuzh ∴ )(zh .2

n

qnqnqn

q zbaz

Silverman and Berman [62] have defined the class ),(*1 BAT and obtained some interesting

results. In this chapter we have obtained some results of Hadamard product (or Convolution)

(Convolution) ∀ ),(*1 BAT and 1( , ),C A B

Lemma 5.1.1 A function ,0,)(2

nqn

qnqn

q azazzf is in ),(*1 BAT if & only if

.1)1()1(

)1()1)((

2

nqna

AqB

ABqn

Lemma 5.1.2 Let .zf(z)2

q

n

qnqn za Where 0qna is in ),(1 BAC if & only if

.1

)1()1(

)1()1)(()(

22

n AqB

ABqnqn

In Chapter 6, we have worked on a subclass of Univalent (or schlicht i. e. a single valued

function) as follows

2

11)(

n

nn zazzf , .01 na

We obtained some results of Convolutions with negative coefficients. We have also obtained the

Coefficient bounds and discuss for sharpness of the results obtained

6.1 Sub classes of univalent functions with negative coefficients

Let S be the class of multivalent normalized functions as given below

2

)(n

nn zazzf ; .0na

Which are Holomorphic (a regular) being in = ; 1 .z z Let T be the subclass of S which is

given as follows

2

( ) ;nn

n

f z z a z

.0na

The properties of these functions were studied by Silverman and Berman [62]. Let K= w;

Holomorphic (a regular) in E, w (0) = 0; )(zw < 1 in E. Also let ( , ) represents the class of

the Holomorphic (an analytic) function in the unit disk which are of the form

,)(1

)(1

zBw

zAw

11 BA .

Here w (z) contained in K. Now we are going to study another subclass of T and we denote it by

1T , which consists

2

11)(

n

nn zazzf ; .01 na

Define .),(&;),('

*

BAGf

zfSffBAS

And .),()(

&;),('

''

BAGf

zfSffBAH

Further we shall define .),(&:),('

1*

1

BAGf

fzHffBAT

And

),()'(

&\),('

'

11 BAGf

zfHffBAC

2

11)(

n

nn zazzf And .)(

2

11

n

nn zbzzg

With .0,0 11 nn ba Their Convolution is defined by )(*)()( zgzfzh

,)(2

111

n

nnn zbazzh .0,0 11 nn ba

Silverman and Berman [62] have defined the class ),(*1 BAT and obtained some interesting

results. In this chapter we have obtained some results of Convolution for the class ),(*1 BAT and

),(1 BAC .

Lemma 6.1.1 A function

2

11)(

n

nn zazzf , .01 na

It is in T1*(A, B) if & only if

.1)()1(

21

nna

AB

ABBn

Lemma 6.1.2 A function

.z(z)2

11

n

nn zaf

Where 01 na is in C1 (A, B) if & only if

.1

)()1()1(1

2

nn

aAB

ABBnn

In Chapter 7, we have considered functions in ∗ ℎ ℎ collection of all z suchthat 0 < | | < 1 = \0, ∀ ∈ & ∈ ∗\0. By using the differential subordi- nation we

derived some certain properties of Meromorphic (analytic except for isolated singularities i. e.

poles) Univalent (or schlicht i. e. a single valued function) functions.

7.1 Definitions and Preliminary Results

Let S gives class of functions defined as( ) = + ∑ ,Where function is univalent (or schlicht i. e. a single valued function) and Holomorphic (an

analytic) in the punctured unit disk ∗, where∗ = collection of all z s. t. 0 < | | < 1 = \0, ∀ ∈ & ∈ ∗\0.

Let , i. e. here after it is to be taken as is sub ordinate to in that

is ( ) ≺ ( ), if ∃ an Holomorphic (an analytic) function ℎ( ) in E, implies that|ℎ( ) | ≤ | | and ( ) = [ℎ( )], ∀ ∈ .

Let is univalent (or schlicht i. e. a single valued function) in E i. e. here after it is to be taken

as the subordination is given as following.( ) ≺ ( )( )⟺ (0) = (0) and ( ) ⊂ ( ).

Let ( ) = 1 + + +⋯⋯ be Holomorphic (an analytic) in , where − ≤ < ≤.s. t. ( ) ≺ ( ∈ ).If & only if ( ) − < ( ),(−1 ≤ 2 < 2 ≤ 1).( ) > , (2 = −1, ∈ ).Several authors like Ali [88], Cho [78], Liu [38], Wang [68], Suffridge [126] and Bulboaca [18]

recently proved some interesting properties of Meromorphic (analytic except for isolated

singularities i. e. poles) Univalent (or schlicht i. e. a single valued function) functions. In this

chapter, we have proved some subordination properties for the class S. When( ) = ( ) = + ∑ ,The convolution ℎ is as defined below.( ∗ )( ) = ( ) = + ∑

Where ( ∈ = ∪ 0, ∈ ). We define a linear Operator by( , , , , , ) ( ) = + ∑ ( )( )( ) + 1 = , , ,, ∗ ( ).Where,

, , , , ,, ( ) = + ∑ [ (1 + ) + 1]&

> 0, ∈ = ∪ 0, ∈ , ≥ 0, ≥ 0,0 < ≤ , 0 < ≤ , > 0, ≥ 0, > .for simplicity throughout the paper we are using( , , , , , ) ( ) = Ω ( ).And = ( )( ) ≠ 0.It is simple to check that( + ) [ ( , , , ) ( )] = Ω ( ) − [ ( + ) + ]Ω ( ).We note that ( , , , , , ) ( ) = ( )And, 1,1,1, , 1 ( ) = ( ) = 2 ( ) + ( ).The above operators satisfies the inequality condition given below

Re ( ) > 0, ∀ . for all ∈ , = exp ( 2 ),

, [ , , , , , ; ] = ∑ , [Ω ( )] = +∙∙∙∙ ( ∈ )., [ , , , , , ; ] = [Ω ( )] + Ω ( ) = +∙∙∙∙ ( ∈ ).ℎ , [ , , , , , ; ] = [Ω ( )] + Ω (− ) = +∙∙∙∙ (ℎ ∈ ).= 1 we have, , [ , , , , , ; ] = Ω ( ).

We introduced the sub classes like , (∝; , , , , , ; ), (∝; , , , , , ; ), (∝; , , , , , ; ), , (∝; , , , , , ; ), , (∝; , , , , , ; ) and

, (∝; , , , , , ; ) of class , these sub classes are containing Meromorphic (analytic

except for isolated singularities i. e. poles) univalent (or schlicht i. e. a single valued function)

functions. Properties of these sub classes like inclusion relationships, integral representation,

convolution properties etc. are studied and derived.

In Chapter 8, we have discussed and verified Convolution and Quasi Convolution properties of

an analytic univalent (or schlicht i. e. a single valued function) and multivalent functions with

+ve & -ve Taylor series expansion.

Preliminary Results

Convolution word arises fromℎ = ∫ ( ) , < 1.i. e. ℓ( ) = ∑ = (1 − )acts as the identity element under Convolution ∗ ℓ = . A many more literature on Diff.

Sub ordination is available in nuture [31], [74], [34], [90] and [121] ,[131], [13], [16], [127] and

[76] etc.

Assuming ( + ) + = , ( ) = , ( + ) = 2Where, ≥ 0, ≥ 0, ≥ 0, 0 ≤ ≤ 1, ( )( ) = , ≥ 0, ≥ 0,≥ 0,0 < ≤ , 0 ≤ < 1, ≥ 0, 0 < ≤ , − ≤ < ≤ .8.2 Preliminary Lemma

Lemma 8.2.1 let ( ) be as given below( ) = − ∑ .I. e. here after it is to be taken as (z) contained in ∗( , , , , , , , , ) if & only if∑ [( ) ( ∝) ( ∝) ]⟨ ( ) ( )⟩ ⟨ [ ( ) ( )]⟩( )( ∝)≤ 1.8.3 Applications of Differential Subordination

Let ( , 1) represents the class of Holomorphic (an analytic) functions in the open unit disk as

given below

( ) = + ∑ (∀ ≥ 0; ∈ ).= : | | < 1 Let ( ), ( ) ∈ ( , 1),Where, ( ) = + ∑And ( ) = + ∑ ,The convolution of above defined functions is as given below.( ∗ )( ) = + ∑Let , , , , and , , be fixed real numbers. A function ( , 1) contained

in the class , , , , , )( ; , ) if it satisfiesℓ , , , ( ) ≺ (z ∈ U)ℓ , , , ( ) = [1 − ( + )] ( ) + ( + ) ( ) ( )Where, ( ) = + ∑ Г )Г ! .( ) = ( )– ( − ) ( ).

Work is due to the motivation by [59], [60] & [49].