Higher dimensional generalizations of the Bloch constant and their lower bounds

Abstract:

A higher dimensional generalization of the classical Bloch theorem depends in an essential way on the “boundedness” of the family of holomorphic mappings considered. In this paper the author considers two types of such “bounded” families and obtains explicit lower bounds of the generalized Bloch constants of these families on the hyperball in the space ${{\mathbf {C}}^n}$ in terms of universal constants which characterize the families.