On some classes of analytic functions

Let m 1 , m 2 be any numbers and let V m 1 , m 2 be the class of functions of analytic in the unit disc E = { z : | z | < 1 } for which f ′ ( z ) = ( S ′ 1 ( z ) ) m 1 ( S ′ 2 ( z ) ) m 2 where S 1 and S 2 are analytic in E with S ′ 1 ( 0 ) = ( S ′ 2 ( 0 ) ) = 1 . Moulis [1] gave a sufficient condition and a necessary condition on parameters m 1 and m 2 for the class V m 1 , m 2 to consist of univalent functions if S 1 and S 2 are taken to be convex univalent functions in E . In fact he proved that if f ϵ V m 1 , m 2 where S 1 and S 2 are convex and m 1 = k + 2 4 e − i α ( 1 − ρ ) cos α , m 2 = k − 2 4 e − i α ( 1 − ρ ) cos α , 2 | m 1 + m 2 | ≤ 1 , then f is univalent in E . In this paper we consider the class V m 1 , m 2 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.