Estimates for generalized Bohr radii in one and higher dimensions

In this article, we study a generalized Bohr radius R-p,R-q(X), p, q is an element of [1, infinity) defined for a complex Banach space X. In particular, we determine the exact value of R-p,R-q(C) for the cases (i) p, q is an element of [1, 2], (ii) p is an element of (2, infinity), q is an element of [1, 2], and (iii) p, q is an element of [2, infinity). Moreover, we consider an n-variable version R-p,q(n)(X) of the quantity R-p,R-q(X) and determine (i) R-p,q(n)(H) for an infinite-dimensional complex Hilbert space 7-C and (ii) the precise asymptotic value of R-p,q(n)(X) as n -> infinity for finite-dimensional X. We also study the multidimensional analog of a related concept called the p-Bohr radius. To be specific, 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.