Khalida I. Noor and Haila Madifer

Copyright © 1987 Khalida I. Noor and Haila Madifer. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

In this paper we consider the class Vm1,m2 in more general way and show that it contains the class of functions with bounded boundary rotation and many other classes related with it. Some coefficient results, arclength problem, radius of convexity and other problems are proved for certain cases. Our results generalize many previously known ones.