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 Fréchet-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 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 are obtained, and the corresponding results of the above generalized mappings on the unit polydisk in C are got as well. Thus, some known results in prior literatures are generalized. [ABSTRACT FROM AUTHOR]