Analytic and mapping properties of certain analytic functions with applications

Abstract

We consider basic hypergeometric functions introduced by Heine. We study mapping properties of certain ratios of basic hypergeometric functions having shifted parameters and show that they map the domains of analyticity onto domains convex in the direction of the imaginary axis. In order to investigate these mapping properties, some useful identities are obtained in terms of basic hypergeometric functions. In addition, we find conditions under which the basic hypergeometric functions are in a q-close-to-convex family. For every q ∈ (0, 1) and 0 ≤ α < 1 we define a class of analytic functions, the so-called q-starlike functions of order α, on the open unit disk. We study this class of functions and explore some inclusion properties with the well-known class of starlike functions of order α. We discuss the Herglotz representation formula for analytic functions zf0 (z)/f(z) when f(z) is q-starlike of order α. As an application we also discuss the Bieberbach conjecture problem for the q-starlike functions of order α. We consider certain subfamilies, of the family of univalent functions in the open unit disk, defined by means of sufficient coefficient conditions for univalency. In this thesis, we study the problem of the well-known conjecture of Zalcman consisting of a generalizedcoefficient functional, the so-called generalized Zalcman conjecture problem, for functions belonging to those subfamilies. We estimate the bounds associated with the generalized coefficient functional and show that the estimates are sharp. we study some necessary conditions for bounded John domains associated with functions in Nehari-type classes. The series of preparatory results, which are applications of certain initial value problems, consist of sharp estimations of pre-Schwarzian derivatives of functions belonging to the Nehari-type classes. In the sequel, we also see that a solution of a complex differential equation has a special form in terms of ratio of hypergeometric functions resulting to an integral representation. Finally, we attempt to study univalent functions f in the unit disk D such that f(D) are unbounded John domains and state some related open problems. We consider the class of all analytic and locally univalent functions f of the form f(z) = z + P∞ n=2 a2n−1z 2n−1 , |z| < 1, satisfying the condition Re 1 + zf00(z) f 0 (z)

> − 1 2 . We show that every section s2n−1(z) = z + Pn k=2 a2k−1z 2k−1 , of f, is convex in the disk |z| < √ 2/3. We also prove that the radius √ 2/3 is best possible, i.e. the number √ 2/3 cannot be replaced by a larger one