The p-Bohr radius of a Banach space

Abstract

Following the scalar-valued case considered by Djakow and Ramanujan (A remark on Bohr’s theorem and its generalizations 14:175–178, 2000) we introduce, for each complex Banach space X and each 1≤?<∞ , the p-Bohr radius of X as the value ??(?)=sup{?≥0:∑?=0∞‖??‖????≤sup|?|<1‖?(?)‖?} where ??∈? for each ?∈ℕ∪{0} and ?(?)=∑∞?=0????∈?∞(?,?) . We show that a complex (possibly infinite dimensional) Banach space X is p-uniformly ℂ -convex for ?≥2 if and only if ??(?)>0 . We study the p-Bohr radius of the Lebesgue spaces ??(?) for different values of p and q. In particular we show that ??(??(?))=0 whenever ?<2 and ???(??(?))≥2 and ??(??(?))=1 whenever ?≥2 and ?′≤?≤? . We also provide some lower estimates for ?2(??(?)) for the values 1≤?<2 .