Geodesic
Invertibility Threshold for Nevanlinna Quotient Algebras
Canadian journal of mathematics, Tome 75 (2023) no. 1, pp. 225-244

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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. 
Nicolau, Artur; Thomas, Pascal J. Invertibility Threshold for Nevanlinna Quotient Algebras. Canadian journal of mathematics, Tome 75 (2023) no. 1, pp. 225-244. doi: 10.4153/S0008414X21000511
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     title = {Invertibility {Threshold} for {Nevanlinna} {Quotient} {Algebras}},
     journal = {Canadian journal of mathematics},
     pages = {225--244},
     year = {2023},
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