Bohr radius for Banach spaces on simply connected domains

2021/11/21 by Vasudevarao Allu, Allu, Vasudevarao, Himadri Halder +1 · 3 citations
Mathematics · #30B10 #30C20 #30C65 #46B20 #46E40 #47A56 #47A63 #Advanced Banach Space Theory #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Rings, Modules, and Algebras

paper · doi:10.48550/arxiv.2111.10880

openalex publication_date 2021/11/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let H^∞(Ω,X) be the space of bounded analytic functions f(z)=∑_n=0^∞ x_nzⁿ from a proper simply connected domain Ω containing the unit disk \mathbbD:=\z∈ ℂ:|z|<1\ into a complex Banach space X with \normf_H^∞(Ω,X) ≤ 1. Let ϕ=\ϕ_n(r)\_n=0^∞ with ϕ₀(r)≤ 1 such that ∑_n=0^∞ ϕ_n(r) converges locally uniformly with respect to r ∈ [0,1). For 1≤ p,q<∞, we denote R_p,q,ϕ(f,Ω,X)= sup \r ≥ 0: \normx₀^p ϕ₀(r) + (∑_n=1^∞ \normx_nϕ_n(r))^q ≤ ϕ₀(r)\ and define the Bohr radius associated with ϕ by R_p,q,ϕ(Ω,X)=inf \R_p,q,ϕ(f,Ω,X): \normf_H^∞(Ω,X) ≤ 1\. In this article, we extensively study the Bohr radius R_p,q,ϕ(Ω,X), when X is an arbitrary Banach space and X is certain Hilbert space. Furthermore, we establish the Bohr inequality for the operator-valued Cesáro operator and Bernardi operator.

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• Arithmetic Bohr radius and Local Banach space theory
• Bohr phenomena for slice regular functions over Quaternions
• The Bohr inequality for vector-valued holomorphic functions with lacunary series in complex Banach spaces

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