Sharp growth and distortion theorems for a subclass of biholomorphic mappings

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