We extend the concept of Bloch
functions and Bloch norm in one complex variable to holomorphic functions defined
in the unit ball 𝔹 of ℂn with values in ℂn. It is known that in several complex
variables the Bloch’s Theorem will fail if we do not put some additional restrictions
on functions besides the usual normalization of the derivative at the origin. We shall
show that many important properties for Bloch functions in one complex variable
have analogs for functions in several complex variables. In particular, we
generalize Bonk’s Distortion Theorem. As applications, we give lower and upper
bounds of Bloch constants for various subfamilies of Bloch functions defined in
𝔹.
Mathematical Subject Classification 2000
Primary: 32H02
Secondary: 32A17
Milestones
Received: 15 July 1990
Revised: 20 December 1990
Published: 1 February 1992
Authors
Xiang Yang Liu
Department of Mathematics
University of Cincinnati
Cincinnati OH 45221
United States
