Estimates for the Bohr Radius of a Faber–Green Condenser in the Complex Plane

Lassère, Patrice; Mazzilli, Emmanuel

doi:10.1007/s00365-016-9359-x

Cited by 5 publications

(3 citation statements)

References 18 publications

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“…[3,15]), for uniform algebras (see [28]), for free holomorphic functions (cf. [30]), for a Faber-Green condenser (see [26]), for vector-valued functions (cf. [16,23,24]), for Hardy space functions (see [5]), and also for functions in several variables (see, for example [8,12,21,29]).…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Estimates for generalized Bohr radii in one and higher dimensions

Das¹

2022

Preprint

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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.

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Section: Introduction and The Main Resultsmentioning

confidence: 99%

Estimates for generalized Bohr radii in one and higher dimensions

Das¹

2022

Preprint

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“…[1]), for ordinary and vector-valued Dirichlet series (see, f.i., [3, 15]), for uniform algebras (see [28]), for free holomorphic functions (cf. [30]), for a Faber–Green condenser (see [26]), for vector-valued functions (cf. [17, 23, 24]), for Hardy space functions (see [5]), and for functions in several variables (see, for example, [2, 8, 12, 21, 29]).…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Estimates for generalized Bohr radii in one and higher dimensions

Das

2022

Can. Math. Bull.

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In this article, we study a generalized Bohr radius 𝑅 𝑝,𝑞 (𝑋), 𝑝, 𝑞 ∈ [1, ∞) defined for a complex Banach space 𝑋. In particular, we determine the exact value of 𝑅 𝑝,𝑞 (C) for the cases ). 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. 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.

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“…[3,15]), for ordinary and vector valued Dirichlet series (see for example [4,10]), for a Faber-Green condenser (cf. [20]), and for free holomorphic functions (see [26]). The reader is also suggested to look at the references of the aforementioned articles to get a clearer picture of the recent developments on this subject.…”

Section: Introductionmentioning

confidence: 99%