2022/04/27 by Nilanjan Das, Das, Nilanjan
Mathematics · #Advanced Banach Space Theory #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2204.12706
openalex publication_date 2022/04/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The generalized Bohr radius Rp, 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 Rp, 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 Rp, qn(X) of the quantity Rp, q(X) and determine (i) Rp, qn(H) for an infinite dimensional complex Hilbert space H, (ii) the precise asymptotic value of Rp, qn(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 Rp, q(Y, X) of Rp, 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 Rp, q(Y, X) is governed by the geometry of X alone.