On the radius of univalence and convexity for some classes of analytic functions

Abstract

In this thesis, we consider the problem of finding the radius of univalence for functions f(z) = z+a_(2)z^(2)+..., which are analytic and satisfy Re {f(z)/g(z)}>0 for │z│<1, where g(z) = z+b_(2)z^(2)+... is analytic and univalent for │z│<1. By observing Macgregor's paper, we determine the largest circle │z│<r such that each functions satisfying Re {zf'(z)/f(z)}>½ for │z│<1 is convex in │z│<r. Moreover, we investigate a class of functions f(z) which satisfy │f(z)/g(z) -1│<1 for │z│<1 where f(z) = z+a_(2)z^(2)+.. is analytic for │z│<1, and g(z) = z+b_(2)z^(2)+... is analytic and univalent for │z│<1. We obtain the following result. 1. If g(z) is starlike and │f(z)/g(z) -1│<1 for │z│<1, then f(z) is univalent and stralike for │z│<1/3. 2. If g(z) is convex and │f(z)/g(z) -1│<1 for │z│<1, then f(z) is univalent and starlike for │z│<√2-1.