Mixed B\"ohr radius in several variables

DANIEL GALICER; MARTIN MANSILLA; SANTIAGO MURO

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY

AMER MATHEMATICAL SOC

Lugar: Providence; Año: 2019

0002-9947

Let $K(B_{ell_p^n},B_{ell_q^n}) $ be the $n$-dimensional $(p,q)$-Bohr radius for holomorphic functions on $CC^n$. That is, $K(B_{ell_p^n},B_{ell_q^n}) $ denotes the greatest number $rgeq 0$ 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$$sup_{z in r cdot B_{ell_q^n}} sum_{alpha} ert a_{alpha} z^{alpha} ert leq sup_{z in B_{ell_p^n}} ert f(z) ert,$$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.