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

In this article, first, the sharp estimates of all homogeneous expansions for a subclass of quasi-convex mappings of type BB 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 CCn 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 AA and quasi-convex mappings of type BB) in several complex variables are get accordingly. Our results state that a weak version of the Bieberbach conjecture for quasi-convex mappings of type BB and order a in several complex variables is proved, and the derived conclusions are the generalization of the classical results in one complex variable.