Sharp estimates of all homogeneous expansions for a subclass of quasi-convex mappings of type B and order α in several complex variables

Abstract In this article, first, the sharp estimates of all homogeneous expansions for a subclass of quasi-convex mappings of type B and order a on the unit ball in complex Banach spaces are given. Second, the sharp estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in C n are also established. In particular, the sharp estimates of all homogeneous expansions for a subclass of quasi-convex mappings (include quasi-convex mappings of type A and quasi-convex mappings of type B ) in several complex variables are get accordingly. Our results state that a weak version of the Bieberbach conjecture for quasi-convex mappings of type B and order a in several complex variables is proved, and the derived conclusions are the generalization of the classical results in one complex variable.