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 number $r\geq 0$ such that for every entire function $f(z)=\sum _{\alpha } a_{\alpha } z^{\alpha }$ in $n$-complex variables, we have the following (mixed) Bohr-type inequality: \begin{equation*} \sup _{z \in r \cdot B_{\ell _q^n}} \sum _{\alpha } \vert a_{\alpha } z^{\alpha } \vert \leq \sup _{z \in B_{\ell _p^n}} \vert f(z) \vert , \end{equation*} 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.