On multidimensional Bohr radii for Banach spaces

TL;DR

This paper investigates the multidimensional Bohr radii for holomorphic functions on unit balls of ll_q spaces with values in Banach spaces, providing asymptotic estimates and applications.

Contribution

It extends the concept of multidimensional Bohr radii to Banach space-valued functions and derives exact asymptotic estimates for finite and infinite dimensions.

Findings

Obtained exact asymptotic estimates of multidimensional Bohr radius.

Derived lower bounds for the arithmetic Bohr radius.

Extended the theory to Banach space-valued holomorphic functions.

Abstract

In this paper, we study a more general version of multidimensional Bohr radii for the holomorphic functions defined on unit ball of $\ell^n_q\,\,(1\leq q\leq \infty)$ spaces with values in arbitrary complex Banach spaces. More precisely, we study the multidimensional Bohr radii for bounded linear operators between complex Banach spaces, primarily motivated by the work of A. Defant, M. Maestre, and U. Schwarting [Adv. Math. 231 (2012), pp. 2837--2857]. We obtain the exact asymptotic estimates of multidimensional Bohr radius for both finite and infinite dimensional Banach spaces. As an application, we find the lower bound of arithmetic Bohr radius.