Mixed Bohr radius in several variables

MURO, SANTIAGO; GALICER, DANIEL; MANSILLA, MARTÍN

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY

AMER MATHEMATICAL SOC

Año: 2020 vol. 373 p. 777 - 796

0002-9947

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 $ rgeq 0$ such that for every entire function $ f(z)=sum _{alpha } a_{alpha } z^{alpha }$ in $ n$-complex variables, we have the following (mixed) Bohr-type inequality:$displaystyle 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.