Local Banach Space Theoretic Approach to Bohr's Theorem for Vector Valued Holomorphic and Pluriharmonic Functions

TL;DR

This paper investigates Bohr's theorem for vector valued holomorphic and pluriharmonic functions using local Banach space invariants, establishing positivity and asymptotic behavior of the Bohr radius across various Banach spaces.

Contribution

It introduces a Banach space theoretic framework to analyze Bohr's theorem, including new results on the Bohr radius and a Schwarz-Pick lemma for operator valued pluriharmonic maps.

Findings

Bohr radius is always strictly positive in the studied setting.

Asymptotic behavior of the Bohr radius is characterized in finite and infinite dimensions.

A coefficient-type Schwarz-Pick lemma for operator valued pluriharmonic maps is established.

Abstract

We study Bohr's theorem for vector valued holomorphic and operator valued pluriharmonic functions on complete Reinhardt domains in $\mathbb{C}^n$. Using invariants from local Banach space theory, we show that the associated Bohr radius is always strictly positive and obtain its asymptotic behavior separately in the finite- and infinite-dimensional settings. The framework developed here includes the classical Minkowski-space setting as a special case and applies to a wide class of Banach sequence spaces, including mixed Minkowski, Lorentz, and Orlicz spaces. We further establish a coefficient-type Schwarz-Pick lemma for operator valued pluriharmonic maps on complete Reinhardt domains.