On some classes of analytic functions

Abstract

Let m1, m2 be any numbers and let Vm1,m2 be the class of functions of analytic in the unit disc E={z:|z|<1} for which f′(z)=(S′1(z))m1(S′2(z))m2 where S1 and S2 are analytic in E with S′1(0)=(S′2(0))=1. Moulis [1] gave a sufficient condition and a necessary condition on parameters m1 and m2 for the class Vm1,m2 to consist of univalent functions if S1 and S2 are taken to be convex univalent functions in E. In fact he proved that if f ϵ Vm1,m2 where S1 and S2 are convex and m1=k+24e−iα(1−ρ)cosα, m2=k−24e−iα(1−ρ)cosα, 2|m1+m2|≤1, then f is univalent in E.