Analytic and geometric properties of certain classes of univalent and P-valent functions

Abstract

This thesis deals with univalent functions as well as p-valent (or multivalent) functions de ned on the unit disk D := fz 2 C : jzj < 1g. Let A denote the family of all normalized analytic functions f(z) = z + P1 n=2 anzn in D and S denote the class of all univalent functions f 2 A. A generalization of close-to-convex functions by means of a q-analog of the di erence operator acting on analytic functions is called the q-close-to-convex functions in D. The class of q-close-to-convex functions is denoted by Kq. We determine the several su cient conditions for f(z) = z + P1 n=2 anzn to be in Kq, where the coe cient an are real, non-negative and connected with certain monotone properties. In addition, we prove the Bieberbach-de Branges Theorem for functions in the class Kq. One of the classical problems concerns the class of analytic functions f on D which have nite Dirichlet integral (1; f), where (r; f) = ZZ jzj<r jf0(z)j2 dxdy (0 < r 1): Computing (r; f) is known as the area problem for the function of type f. The class S (A;B) of functions f 2 A and satis es the subordination condition zf0(z)=f(z) (1 + Az)=(1 + Bz) in D and for some 􀀀1 B 0, A 2 C with A 6= B, has been studied extensively. We are mainly interested to discuss the extremal problem of determiningthe value of max f2S (A;B) (r; z=f) as a function of r. This settles the question raised by Ponnusamy and Wirths (Ann. Acad. Sci. AI. Math. 39:721-731, 2014). The class of analytic p-valent functions f(z) = zp+ P1 n=p+1 anzn; p 2 N is denoted by Ap. For f 2 Ap, let us consider the integral means L(r; f; p) = r2p 2 Z 􀀀 d jf(rei )j2 ; r 2 (0; 1): We also focus on computing the integral means and the analog of area problems for certain subclasses of p-valent functions. We estimate the Taylor-Maclaurin coe cients of functions belonging to related p-valent functions. These estimation improve the results of Aouf [7, 8]. We introduce a new class, denoted by Pa;b;c( ), in terms of convolution ( ) with Gaussian hypergeometric functions 2F1(a; b; c; z), which is de ned by Pa;b;c( ) =

f 2 A : f(z) z = 2F1(a; b; c; z) p(z); a b < c

; where p is an analytic function with positive real part of order (0 < 1) in D and p(0) = 1. Making use of duality principle, we investigate the order of starlikeness (or convexity) of the integral transform V (f)(z) = R 1 0 (t) f(tz) t dt over functions f in the class Pa;b;c( ).