GEOMETRIC PROPERTIES OF INTEGRAL TRANSFORMS OF A SECOND ORDER DIFFERENTIAL INEQUALITY

Abstract

Geometric function theory is a branch of classical complex analysis which deals with the study of geometric properties of analytic functions. One of the broader area of current research in geometric function theory is mainly concerned with the theory of univalent functions. The most important fundamental result from the univalent function theory is the Riemann mapping theorem, which states that every simply connected and proper open subdomain of the complex plane can be mapped conformally onto the open unit disk, upon an univalent function. Let A be the class of all analytic functions f in the open unit disk D = fz 2 C : jzj < 1g, which are normalized by the condition f(0) = f0(0) 􀀀 1 = 0, and S, be the subclass of A, consisting of all univalent functions in D. For the two normalized analytic functions f1(z) = z+ X1 n=2 anzn and f2(z) = z+ X1 n=2 bnzn de ned in D, the convolution or Hadamard product, denoted by ` ', is given by (f1 f2)(z) = z + X1 n=2 anbnzn; z 2 D: In order to investigate the geometric properties of the analytic functions, several techniques are available in the literature. Duality technique helps in solving complicated problems very easily and also provides sharp results. For the most fundamental results from the theory of duality that are used in thesis are from the monograph of S. Ruscheweyh [116]. One of the application of this technique for convolution in connection with an integral operator, was implemented by R. Fournier and S. Ruscheweyh in [36]. Let (t) : [0; 1] ! R be a non-negative integrable function which satisfy the normalization condition Z 1 0 (t)dt = 1. For f 2 A, consider the generalized integral transform i de ned by V (f)(z) = Z 1 0 (t)

f(tz) t

dt !1= ; > 0 and z 2 D where z or f represents only the principle part. This non-linear integral operator was considered in the work of A. Ebadian et al. [34]. For the case = 1, this operator was studied by R. Fournier and S. Ruscheweyh [36] and later by several authors, for di erent choices of (t) (see [7, 9, 15, 138]). Note that V (f)(z) : V 1 (f)(z). The analytic characterization of few of the important subclasses of the class S are given below: S ( ) =

f 2 A : Re

zf0(z) f(z)

> ; 0 < 1; z 2 D

; C( ) =

f 2 A : Re

1 + zf00(z) f0(z)

> ; 0 < 1; z 2 D

; K( ) =

f 2A : Re

ei zf0(z) g(z)

> ; 2 R; 0 < 1; g 2 S and z 2 D

and C ( ) =

f 2 A : Re

1 􀀀 1

zf0(z) f(z)

+ 1

1 + zf00(z) f0(z)

> ; > 0; 0 < 1; z 2 D o ; where S ( ), C( ), K( ) and C ( ) are the class of starlike functions, convex functions, close-to-convex functions and 1= - convex functions of order , respectively. Note that S (0) S , C(0) C and K(0) K, respectively are the well known classes of starlike functions, convex functions and close-to-convex functions that have rich geometric properties. The Pascu class denoted byM( ; ), 0 < 1, and de ned in D, is the hybridization of two important subclasses: the class of starlike functions and the class of convex functions of S. Analytically, this class is de ned as M( ; ) =

f 2 A : Re z(zf0(z))0 + (1 􀀀 )zf0(z) zf0(z) + (1 􀀀 )f(z) > ; 0 1; z 2 D

: ii Note that M(0; ) M( ). One of the main objective of the thesis is to generalize and unify both the function class P ( ; ), de ned in D and is given as P ( ; )= ( f 2A : Re ei

(1􀀀 )

f z

+

f z

zf0 f

􀀀 ! > 0; 2 R ) and the integral operator V explained before, so as to de ne new functionals involving the admissible functions and study the univalency and other geometric properties, using duality technique. For this purpose, a more generalized class of P ( ; ) is de ned in this thesis as under: For 0, 0 < 1, > 0 and 0, let W ( ; ) = ( f 2A : Re ei

(1􀀀 +2 )

f z

+

􀀀3 +

1􀀀 1

zf0 f

+ 1

1+ zf00 f0

f z

zf0 f

􀀀 ! > 0; z 2 D; 2 R ) : Using this class and the integral operator V (f)(z), the objective of the thesis is addressed with several interesting applications. In the course of this study, interesting analysis are made to study various consequences of W ( ; ) : W1 ( ; ) as well. The summary of all the seven chapters incorporated in this thesis is as follows. Chapter 1 consist of fundamental de nitions, elementary concepts and few of the existing literature on the geometric function theory that are used in the sequel. This chapter constitutes the structures for the remaining chapters. In Chapter 2, the conditions between the parameter and the function (t) are obtained, under which the integral operator V maps the function from the analytic class W ( ; ) into the class M( ) : M(0; ). For particular choices of (t), interesting applications related to various well-known integral operators are obtained. The study is also explored with the generalized linear operator, the operator formed with the convex combination of z and V , which was de ned by R. M. Ali and V. Singh [9]. These results are further extended for the class M( ; ) and are explained in Chapter 3. iii The admissible and su cient conditions on (t) are examined in Chapter 3, so that the integral transforms V carries the function from the class W ( ; ) to M( ; ). Various applications related to known integral operators are discussed. Clear analysis is provided to explain the di erences and similarities of the results obtained in Chapter 2 and particular cases of Chapter 3. The study of convex hull and extreme points for the class W (1; 0) was pursued by D. J. Hallenbeck [49], and later H. Silverman [124] analysed for the analytic functions class W (3; 1); and more recently, in [39, 74, 141]. De nition 1. An extreme point of a set E is a point of E that cannot be written as a proper convex combination of any two points of E. Chapter 4 provides the extreme points and sharp coe cient bounds for the class W ( ; ). We also nd the estimates on , that would ensure functions in W ( ; ) are starlike. When = 0, a sharp radius of univalence is obtained for the class W0( ; ). De nition 2. The Schwarzian derivative (Sf (z)) of the analytic and locally univalent function f in the domain D is de ned by the expression Sf (z) =

f00(z) f0(z) 0 􀀀 1 2

f00(z) f0(z) 2 : It acts as an important tool in obtaining the su cient condition for univalence for the locally univalent analytic function. The su cient conditions in terms of the Schwarzian derivative and the second Taylor coe cient is attained in Chapter 4 for functions belonging to the class W0( ; ). In Chapter 5, the necessary and su cient condition is presented for starlikeness of the generalized and non-linear integral transforms V for the function belonging to the analytic functions class W ( ; ). Applications associated with the well-known integral operators are also discussed. Chapter 6 deals with the admissible and su cient conditions on the parameter and the function (t) so that the generalized and non-linear integral operator V can be mapped to the function from the class W ( ; ) into C ( ). Interesting consequences corresponding to the known integral operators are examined. iv Chapter 7 determines the conditions between the parameters 1, 1, 1, 2, 2 and 2, that carries the function from the class W 1( 1; 1) into W 2( 2; 2) for the generalized and non-linear integral operator V . We also study the number of applications for speci c choices of (t) and the comparison is made with the existing results. v