Sharp growth and distortion theorems for a subclass of biholomorphic mappings

Abstract

Let X be a complex Banach space with norm ‖⋅‖, and B be the unit ball in X. In this paper, we introduce a class of holomorphic mappings Mg on B. Let F(x) be a normalized locally biholomorphic mapping on B such that (DF(x))−1F(x)∈Mg. We investigate the sharp growth theorem for F(x). As applications, the sharp distortion theorems for a subclass of starlike mappings are obtained.