# Invertibility Threshold for Nevanlinna Quotient Algebras

## TL;DR

This paper characterizes the invertibility conditions in Nevanlinna quotient algebras based on harmonic majorants and explores similar conditions in Smirnov class quotients, advancing understanding of function invertibility in complex analysis.

## Contribution

It provides a new criterion for invertibility in Nevanlinna quotient algebras involving harmonic majorants and extends the analysis to Smirnov class quotients.

## Key findings

- Invertibility criterion linked to harmonic majorants on specific sets
- Characterization of harmonic functions H satisfying the invertibility condition
- Extension of results to quotients of the Smirnov class

## Abstract

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 $H$ in the unit disc such that the latter condition holds. We also discuss the analogous invertibility problem in quotients of the Smirnov class.