Decrease of bounded holomorphic functions along discrete sets

2001/06/07 by Jordi Pau, Pau, Jordi, Pascal J. Thomas +1
Computer Science · Engineering · Mathematics · #30D50 (Primary) #30D55 (Secondary) #Aerospace Engineering and Control Systems #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Mathematical Control Systems and Analysis #advanced mathematical theories #math.CA #math.CV #msc:30D50 #msc:30D55

paper · pdf · doi:10.48550/arxiv.math/0106048

13 pp ; rewritten introduction, proofs revised for clarity, a number of minor corrections

openalex publication_date 2001/06/07 · arxiv created 2002/09/06 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We provide results of uniqueness for holomorphic functions in the Nevanlinna class bridging those previously obtained by Hayman and Lyubarskii-Seip. Namely, we propose certain classes of hyperbolically separated sequences in the disk, in terms of the rate of non-tangential accumulation to the boundary (the endpoints of this spectrum of classes being respectively the sequences with a non-tangential cluster set of positive measure, and the sequences violating the Blaschke condition); and for each of those classes, we give a critical condition of radial decrease on the modulus which will force a Nevanlinna class function to vanish identically.

Related

• Nevanlinna classes associated to a closed set on ∂D
• Equivalence of summatory conditions along sequences for bounded holomorphic functions
• Sampling Sets for the Nevanlinna class
• Non-invariance of the Nevanlinna class under the action of the Cesaro and Volterra operators
• Hyperbolic distances, nonvanishing holomorphic functions and Krzyz's conjecture