The Bohr radius of a Banach space

Abstract. Let 1 ≤ p, q < ∞ and let X be a complex Banach space. For each f(z) = P∞ n=0 xnz n with ‖f‖H∞(D,X) ≤ 1 we define Rp,q(f,X) = sup{r ≥ 0: ‖x0‖p+(P∞n=1 ‖xn‖rn)q ≤ 1} and denote the Bohr radius of X by Rp,q(X) = inf{Rp,q(f,X) : ‖f‖H∞(D,X) ≤ 1}. The aim of this note is to study for which spaces X = Ls(µ) or X = `s one has Rp,q(X)> 0