ON BIHOLOMORPHIC MAPPINGS IN COMPLEX BANACH SPACES

Let X be a complex Banach space with norm parallel to center dot parallel to, B the unit ball in X. In this paper, we introduce a class of holomorphic mappings M(g) on B. Let f(x) be a normalized locally biholomorphic mapping on B such that (Df(x))(-1) f (x) is an element of M(g). We obtain the growth and covering theorems, as well as coefficient estimates for f (x). Especially, as corollaries, we unify and generalize many known results. Moreover, in view of proofs of corollaries, we can see the relations among the subclasses of starlike mappings.