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

Abstract

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 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.

Keywords

Distortion theorem / A zero of order k + 1 / Fréchet-derivative / Jacobi determinant / Parametric representation

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