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On growth and covering theorems of quasi-convex mappings in the unit ball of a complex banach space

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.