Invertibility Threshold for Nevanlinna Quotient Algebras

Let $\mathcal {N}$ be the Nevanlinna class, and let B be a Blaschke product. It is shown that the natural invertibility criterion in the quotient algebra $\mathcal {N} / B \mathcal {N}$ , that is, $|f| \ge e^{-H} $ on the set $B^{-1}\{0\}$ for some positive harmonic function H, holds if and only if...

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Bibliographic Details
Published inCanadian journal of mathematics Vol. 75; no. 1; pp. 225 - 244
Main Authors Nicolau, Artur, Thomas, Pascal J.
Format Journal Article
LanguageEnglish
Published Canada Canadian Mathematical Society 01.02.2023
Cambridge University Press
University of Toronto Press
Subjects
Algebra
Complex Variables
Harmonic functions
Mathematics
Quotients
30J10
30H80
30H15
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Summary:Let $\mathcal {N}$ be the Nevanlinna class, and let B be a Blaschke product. It is shown that the natural invertibility criterion in the quotient algebra $\mathcal {N} / B \mathcal {N}$ , that is, $|f| \ge e^{-H} $ on the set $B^{-1}\{0\}$ for some positive harmonic function H, holds if and only if the function $- \log |B|$ has a harmonic majorant on the set $\{z\in \mathbb {D}:\rho (z,\Lambda )\geq e^{-H(z)}\}$ , at least for large enough functions H. We also study the corresponding class of positive harmonic functions H on the unit disc such that the latter condition holds. We also discuss the analogous invertibility problem in quotients of the Smirnov class.
ISSN:0008-414X
1496-4279
DOI:10.4153/S0008414X21000511