Sharp distortion theorems for a subclass of biholomorphic mappings which have a parametric representation in several complex variables

In this paper, the sharp distortion theorems of the Frechet-derivative type for a subclass of biholomorphic mappings which have a parametric representation on the unit ball of complex Banach spaces are established, and the corresponding results of the above generalized mappings on the unit polydisk in C n are also given. Meanwhile, the sharp distortion theorems of the Jacobi determinant type for a subclass of biholomorphic mappings which have a parametric representation on the unit ball with an arbitrary norm in C n are obtained, and the corresponding results of the above generalized mappings on the unit polydisk in C n are got as well. Thus, some known results in prior literatures are generalized.