Equivalence of summatory conditions along sequences for bounded holomorphic functions

2003/06/26 by Vladimir Eiderman, Vladimir Ya. Eiderman, Eiderman, Vladimir Ya. +2
Mathematics · #30D50 #Algebraic and Geometric Analysis #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #math.CA #math.CV #msc:30D50

paper · pdf · doi:10.48550/arxiv.math/0306376

15 pages, LaTeX; some typos corrected. To appear in the issue of Complex Variables dedicated to the memory of Matts Essen

openalex publication_date 2003/06/26 · arxiv created 2003/11/24 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A sequence of points z_k in the unit disk is said to be thin for a given decrease function ρ, if there is a nontrivial bounded holomorphic function such that the infinite series ∑_k ρ(1-|z_k|)|f(z_k)| converges. All sequences will be assumed hyperbolically separated. We give necessary and sufficient conditions for the problem of thinness of a sequence to be non-trivial (one way or the other), and for two different decrease functions to give rise to the same thin sequences. Along the way, some concrete conditions (necessary or sufficient) for a sequence to be thin are obtained.

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