H. Bohr's theorem for bounded symmetric domains

TL;DR

This paper generalizes Harald Bohr's classical theorem from the unit disc to all bounded symmetric domains, establishing an optimal bound for the sum of absolute values of Taylor coefficients of holomorphic maps.

Contribution

It extends Bohr's theorem to all bounded symmetric domains with a classification-independent proof, broadening its applicability in complex analysis.

Findings

The Bohr radius is established for all bounded symmetric domains.

The proof does not rely on domain classification.

The bound 1/3 is shown to be optimal.

Abstract

A theorem of Harald Bohr (1914) states that if f is a holomorphic map from the unit disc into itself, then the sum of absolute values of its Taylor expansion is less than 1 for |z|<1/3. The bound 1/3 is optimal. This result has been extended in a suitable sense by Liu Taishun and Wang Jianfei (2007) to the bounded complex symmetric domains of the four classical series, and to polydiscs. The result of Liu and Wang may be generalized to all bounded symmetric domains, with a proof which does not depend on classification.