The Bohr Radius of a Banach Space

Abstract

Let 1≤p, q<∞ and let X be a complex Banach space. For each $f(z) = \sum\nolimits_{n = 0}^\infty {x_n z^n }$ with $\left\| f \right\| _{H\infty (\mathbb{D},X)} \leqslant 1$ we define $R_{p,q} (f, X) = sup\{ r \geqslant 0 : \left\| {x_0 } \right\|^p + (\sum\nolimits_{n = 1}^\infty {\left\| {x_n } \right\|r^n } )^q \leqslant 1\}$ and denote the Bohr radius of X by $R_{p,q} \left( X \right) = \inf \left\{ {R_{p,q} \left( {f,X} \right):\left\| f \right\|_{H\infty (\mathbb{D},X)} \leqslant 1} \right\}.$ The aim of this note is to study for which spaces X=L ^s(μ) or X = ℓ ^s one has R _p,q (X)>0.