Abstract
A class of biholomorphic mappings named quasi-convex mapping is introduced in the unit ball of a complex Banach space. It is proved that this class of mappings is a proper subset of the class of starlike mappings and contains the class of convex mappings properly, and it has the same growth and covering theorems as the convex mappings. Furthermore, when the Banach space is confined to Cn, the quasi-convex mapping is exactly the quasi-convex mapping of type A introduced by K.A. Roper and T. J. Suffridge.
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