The radius of convexity for starlike functions of order 1\over2

Abstract

As Strohhicker has pointed out neither (3) nor (4) are sufficient for the convexity of f(z) in Ilz <1. Indeed, a function may satisfy (4) without being univalent for I zj < 1. In this paper we determine the largest circle I zj <r such that each function satisfying (3) is convex in I zj <r. Also, we find the radius of univalence (and starlikeness) for functions satisfying (4). Some properties of functions subject to (3) have been discussed in [5]. These functions are a particular case of the so-called starlike functions of order p which are defined by the condition