Bloch constant of holomorphic mappings on the unit polydisk of ℂ n

Abstract

In this paper, we give a definition of Bloch mappings defined in the unit polydisk Dn, which generalizes the concept of Bloch functions defined in the unit disk D. It is known that Bloch theorem fails unless we have some restrictive assumption on holomorphic mappings in several complex variables. We shall establish the corresponding distortion theorems for subfamilies β(K) and βloc(K) of Bloch mappings defined in the polydisk Dn, which extend the distortion theorems of Liu and Minda to higher dimensions. As an application, we obtain lower and upper bounds of Bloch constants for various subfamilies of Bloch mappings defined in Dn. In particular, our results reduce to the classical results of Ahlfors and Landau when n = 1.