Estimates for generalized Bohr radii in one and higher dimensions. Issue 2 (4th June 2023)

• Record Type: Journal Article
• Authors: Das, Nilanjan
• Is Part Of: Canadian mathematical bulletin =. Volume 66:Issue 2(2023)
• Page Start: 682
• Page End: 699
• Publication Date: 2023-06-04
• Subjects: 30B10 -- 32A05 -- 46E40 -- 30H05 -- 32A10 -- 32A17 -- 46G20; Bohr radius -- holomorphic functions -- Banach spaces; Mathematics -- Periodicals; Mathematics; Periodicals; 510.5
• DOI: 10.4153/S0008439522000674
• Languages: English
• ISSNs: 0008-4395

Abstract

Abstract: In this article, we study a generalized Bohr radius $R_{p, q}(X), p, q\in [1, \infty )$ defined for a complex Banach space X . In particular, we determine the exact value of $R_{p, q}(\mathbb {C})$ for the cases (i) $p, q\in [1, 2]$, (ii) $p\in (2, \infty ), q\in [1, 2]$, and (iii) $p, q\in [2, \infty )$ . Moreover, we consider an n -variable version $R_{p, q}^n(X)$ of the quantity $R_{p, q}(X)$ and determine (i) $R_{p, q}^n(\mathcal {H})$ for an infinite-dimensional complex Hilbert space $\mathcal {H}$ and (ii) the precise asymptotic value of $R_{p, q}^n(X)$ as $n\to \infty $ 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 .