Estimates for generalized Bohr radii in one and higher dimensions

Abstract

The generalized Bohr radius R p,q (X), p, q ∈ [1, ∞) for a complex Banach space X was introduced by Blasco in 2010. In this article, we determine the exact value of R p,q (C) for the cases (i) p, q ∈ [1, 2], (ii) p ∈ (2, ∞), q ∈ [1, 2] and (iii) p, q ∈ [2, ∞). Moreover, we consider an n-variable version R n p,q (X) of the quantity R p,q (X) and determine (i) R n p,q (H) for an infinite dimensional complex Hilbert space H, (ii) the precise asymptotic value of R n p,q (X) as n → ∞ for finite dimensional X. We also study the multidimensional analogue of a related concept called the p-Bohr radius, introduced by Djakov and Ramanujan in 2000.In particular, we obtain the asymptotic value of the n-dimensional p-Bohr radius for bounded complex-valued functions, and in the vector-valued case we provide a lower estimate for the same, which is independent of n. In a similar vein, we investigate in detail the multidimensional p-Bohr radius problem for functions with positive real part. Towards the end of this article, we pose one more generalization R p,q (Y, X) of R p,q (X)-considering functions that map the open unit ball of another complex Banach space Y inside the unit ball of X, and show that the existence of nonzero R p,q (Y, X) is governed by the geometry of X alone.