On the precise growth, covering, and distortion theorems for normalized biholomorphic mappings

In this paper, we consider a normalized biholomorphic mapping f (x) defined on the unit ball in a complex Banach space, where the origin 0 is a zero of order k + 1 of f (x) − x. The precise growth and covering theorem for f (x) is obtained when f (x) is a starlike mapping or a starlike mapping of order α. Especially, the precise growth and covering theorem for f (x) is also established when f (x) is a quasi-convex mapping. Moreover, the precise distortion theorem for f (x) is given when f (x) is a convex mapping. Our result includes many known results.  2004 Elsevier Inc. All rights reserved