Multidimensional Bohr radii for holomorphic functions with values in complex Banach spaces
Vasudevarao Allu, Himadri Halder, Subhadip Pal
TL;DR
The paper addresses the problem of determining exact multidimensional Bohr radii for holomorphic functions taking values in complex Banach spaces on complete Reinhardt domains. It develops a framework connecting Bohr radii to arithmetic Bohr radii, derives asymptotic estimates for the unit balls of $ℓ^n_q$ and proves the key identity $A_p(B_{ℓ^n_q}, X, λ) = r^n_p(B_{ℓ^n_q}, X, λ)/n^{1/q}$, which in turn yields the exact value $r^n_1(B_{ℓ^n_1}, ℂ, 1) = 1/3$ and corresponding arithmetic constants $A_{p,1}(B_{ℓ^n_1}, ℂ, 1) = p/(n(p+2))$ for $1≤p≤2$. The results connect arithmetic and classical Bohr radii in high dimensions, provide sharp asymptotics for $ℓ^n_q$ balls, and extend Bohr-type phenomena to vector-valued holomorphic functions on Reinhardt domains, with implications for monomial convergence and dimension-dependent behavior.
Abstract
The main aim of this paper is to answer certain open questions related to the exact values of multidimensional Bohr radii by using the concept of arithmetic Bohr radius for holomorphic functions defined in complete Reinhardt domains in $\mathbb{C}^n$ with values in complex Banach spaces. More specifically, for holomorphic functions with values in arbitrary complex Banach spaces, we explore the asymptotic estimates of the arithmetic Bohr radius in the unit ball of $\ell^n_q$ $(1\leq q\leq \infty)$ spaces. As an application, we obtain the exact value of multidimensional Bohr radius for the unit ball in $\ell^n_1$ spaces which improves the previously known result of Aizenberg.