Mixed Bohr radius in several variables

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 $r\ge 0$ 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 $$\underset{z\in r\cdot B_{{\ell }_q^n}}{\sup }\sum _{\alpha }|c_{\alpha }{z}^{\alpha }|\le \underset{z\in B_{{\ell }_p^n}}{\sup }|f(z)|,$$ where $B_{{\ell }_r^n}$ denotes the closed unit ball of the $n$-dimensional sequence space ${\ell }_r^n$. For every $1\le p,q\le \infty$, we exhibit the exact asymptotic growth of the $(p,q)$-Bohr radius as $n$ (the number of variables) goes to infinity.

25 pages