Bloch Constant of Holomorphic Mappings on the Unit Ball of ℂ ⁿ*

Abstract

In this paper, the authors establish distortion theorems for various subfamilies H _k($\Bbb {B}$) of holomorphic mappings defined in the unit ball in ℂ ⁿ with critical points, where k is any positive integer. In particular, the distortion theorem for locally biholomorphic mappings is obtained when k tends to +∞. These distortion theorems give lower bounds on | det f′(z)| and Re det f′(z). As an application of these distortion theorems, the authors give lower and upper bounds of Bloch constants for the subfamilies β _k(M) of holomorphic mappings. Moreover, these distortion theorems are sharp. When $\Bbb {B}$ is the unit disk in ℂ, these theorems reduce to the results of Liu and Minda. A new distortion result of Re det f′(z) for locally biholomorphic mappings is also obtained.

Keywords

Bloch constant / Holomorphic mappings / Locally biholomorphic mappings / Critical points / 32H02 / 32H99

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