An absolute estimate of the homogeneous expansions of holomorphic mappings

Taishun Liu and Jianfei Wang

Vol. 231 (2007), No. 1, 155–166

DOI: 10.2140/pjm.2007.231.155

Abstract

Let f : Ω → Ω be a holomorphic mapping, where Ω is one of the four classical domains in ℂ^m×n. We show that, if P = f(0), we have

for ∥Z∥_Ω < and φ_P ∈ Aut Ω such that φ_P(P) = 0. This generalizes to higher dimensions a classical result of Bohr, which corresponds to the case Ω = {z : |z| < 1}⊂ ℂ. The constant is the best possible.

Keywords

holomorphic mapping, homogeneous expansion, classical domains, Bohr’s theorem

Mathematical Subject Classification 2000

Primary: 30H05

Secondary: 32A05

Milestones

Received: 23 December 2005

Revised: 25 September 2006

Accepted: 26 October 2006

Published: 1 May 2007