Abstract
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.
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