An application of a subordination chain

Abstract

Let K denote the class of functions g(z)=z+a2z2+⋯ which are regular and univalently convex in the unit disc E. In the present note, we prove that if f is regular in E, f(0)=0, then for g∈K, f(z)+αzf′(z) ≺ g(z)+αzg′(z) in E implies that f(z)≺g(z) in E, where α>0 is a real number and the symbol “≺” stands for subordination.