Convex sum of univalent functions

Abstract

Let f(z)=z + … be regular in the unit disc |z| < 1 (hereafter called E). In a recent paper Trimble [7] has proved that if f(z) be convex in E, then F(z) = (1 − λ)z + λf(z) is starlike with respect to the origin in E for (2/3) ≦ λ ≦ 1. The purpose of this note is to show that if certain additional restrictions be imposed on f(z), then F(z) becomes starlike for all λ, 0 ≦ λ ≦ 1. Also we consider some related problems.