Multidimensional Bohr radii for holomorphic functions with values in complex Banach spaces

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.