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.