$$\alpha $$-Bloch Mappings on Bounded Symmetric Domains in $${\mathbb {C}}^n$$

Abstract

Let $${\mathbb {B}}_X$$ be a bounded symmetric domain realized as the open unit ball of a finite dimensional JB*-triple $$X=({\mathbb {C}}^n, \Vert \cdot \Vert _X)$$ . In this paper, we give a definition of $$\alpha $$ -Bloch mappings on $${\mathbb {B}}_X$$ which is a generalization of $$\alpha $$ -Bloch functions on the unit disc in $${\mathbb {C}}$$ . This definition is new in the case of the Euclidean unit ball $${\mathbb {B}}^n$$ in $${\mathbb {C}}^n$$ . We generalize Bonk’s distortion theorem to $$\alpha $$ -Bloch mappings on $${\mathbb {B}}_X$$ . As an application, we give a lower bound of the Bloch constant for $$\alpha $$ -Bloch mappings on $${\mathbb {B}}_X$$ .