Mixed Bohr radius in several variables

TL;DR

This paper investigates the asymptotic behavior of the mixed $(p,q)$-Bohr radius for holomorphic functions in several complex variables, providing exact growth rates as the number of variables increases.

Contribution

It determines the precise asymptotic growth of the $(p,q)$-Bohr radius for all $1 \,\leq\, p,q \leq \infty$ as the dimension tends to infinity.

Findings

Exact asymptotic growth rates of the $(p,q)$-Bohr radius for large $n$.

Unified analysis covering all $p,q$ in the range.

Extension of classical Bohr radius concepts to mixed-variable settings.

Abstract

Let $K(B_{\ell_p^n},B_{\ell_q^n}) $ be the $n$-dimensional $(p,q)$-Bohr radius for holomorphic functions on $\mathbb C^n$. That is, $K(B_{\ell_p^n},B_{\ell_q^n}) $ denotes the greatest constant $r\geq 0$ such that for every entire function $f(z)=\sum_{\alpha} c_{\alpha} z^{\alpha}$ in $n$-complex variables, we have the following (mixed) Bohr-type inequality $$\sup_{z \in r \cdot B_{\ell_q^n}} \sum_{\alpha} | c_{\alpha} z^{\alpha} | \leq \sup_{z \in B_{\ell_p^n}} | f(z) |,$$ where $B_{\ell_r^n}$ denotes the closed unit ball of the $n$-dimensional sequence space $\ell_r^n$. For every $1 \leq p, q \leq \infty$, we exhibit the exact asymptotic growth of the $(p,q)$-Bohr radius as $n$ (the number of variables) goes to infinity.