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

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.