Estimates for generalized Bohr radii in one and higher dimensions

TL;DR

This paper determines exact values and asymptotic behaviors of generalized Bohr radii in one and higher dimensions for various Banach spaces, extending classical results and exploring multidimensional analogues and related concepts.

Contribution

It provides explicit calculations of generalized Bohr radii for complex spaces and their asymptotics, and introduces a new generalization considering mappings between different Banach spaces.

Findings

Exact values of $R_{p, q}(C)$ for specific $p, q$ ranges.

Asymptotic behavior of $R_{p, q}^n(X)$ as $n$ grows.

Asymptotic values of the $n$-dimensional $p$-Bohr radius for bounded functions.

Abstract

The generalized Bohr radius $R_{p, q}(X), p, q\in[1, \infty)$ for a complex Banach space $X$ was introduced by Blasco in 2010. In this article, we determine the exact value of $R_{p, q}(\mathbb{C})$ for the cases (i) $p, q\in[1, 2]$, (ii) $p\in (2, \infty), q\in [1, 2]$ and (iii) $p, q\in [2, \infty)$. Moreover, we consider an $n$-variable version $R_{p, q}^n(X)$ of the quantity $R_{p, q}(X)$ and determine (i) $R_{p, q}^n(\mathcal{H})$ for an infinite dimensional complex Hilbert space $\mathcal{H}$, (ii) the precise asymptotic value of $R_{p, q}^n(X)$ as $n\to\infty$ for finite dimensional $X$. We also study the multidimensional analogue of a related concept called the $p$-Bohr radius, introduced by Djakov and Ramanujan in 2000. In particular, we obtain the asymptotic value of the $n$-dimensional $p$-Bohr radius for bounded complex-valued functions, and in the vector-valued case we provide a lower estimate for the same, which is independent of $n$. In a similar vein, we investigate in detail the multidimensional $p$-Bohr radius problem for functions with positive real part. Towards the end of this article, we pose one more generalization $R_{p, q}(Y, X)$ of $R_{p, q}(X)$-considering functions that map the open unit ball of another complex Banach space $Y$ inside the unit ball of $X$, and show that the existence of nonzero $R_{p, q}(Y, X)$ is governed by the geometry of $X$ alone.