Deficient rational functions and Ahlfors's theory of covering surfaces
type: Item
instance of: scholarly article
title: Deficient rational functions and Ahlfors's theory of covering surfaces
zbMATH Open document ID: 1021.30029
DOI: 10.1007/BF02388796
author: A. Sauer
published in: Arkiv för Matematik
publication date: 2003-06-16T00:00:00Z
review text: Let \(f(z)\) be a transcendental meromorphic function in the plane, and let \(a_k\), \(k=1,\dots, q\) be distinct complex numbers or \(\infty\). The author considers a generalization of \textit{R. Nevanlinna's} second fundamental theorem \[ (q-2)T(r,f)\leq\sum_{k=1}^q\overline{N}\bigg(r,\frac{1}{f-a_k}\bigg)+S(r,f).\leqno{(1)} \] for distinct small functions. \textit{N. Steinmetz } proved [J. Reine Angew. Math. 368, 134-141 (1986; Zbl 0598.30001) ] \[ (q-2-\varepsilon)T(r,f)\leq\sum_{k=1}^q{N}\bigg(r,\frac{1}{f-a_k}\bigg)+S(r,f),\leqno{(2)} \] for distinct small functions, see also \textit{C. F Osgood} [J. Number theory 21, 347-389 (1985; Zbl 0575.10032)], \textit{G. Frank} and \textit{G. Weissenborn} [Bull. Lond. Math. Soc. 18, 29-33 (1986; Zbl 0586.30025 ]. It has been an open question whether (2) holds if \(N\) is replaced by \(\overline{N}\). The author proves that for rational functions \(R_k\), \(k=1,\dots, q\) with distinct values at \(\infty\), \(a_k\) in (1) can be replaced by \(R_k\). The key idea for the proof is the combination of \textit{Alhfors'} second fundamental theorem and \textit{Rouché's} theorem.
reviewed by: Katsuya Ishizaki
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 1928898
zbMATH Keywords: value distribution theory
zbMATH Keywords: Ahlfors theory of covering surfaces
MaRDI profile type: Publication
cites work: On the theory of covering surfaces.
cites work: Rational Deficient Functions of Meromorphic Functions
cites work: Uniqueness of entire functions sharing four small functions
cites work: A Note on Ahlfors' Theory of Covering Surfaces
cites work: Sometimes effective Thue-Siegel-Roth-Schmidt-Nevanlinna bounds, or better
cites work: Eine Verallgemeinerung des zweiten Nevanlinnaschen Hauptsatzes.
cites work: Some generalizations of the unicity theory of Nevanlinna
cites work: Q4274978
full work available at URL: https://doi.org/10.1007/bf02388796
OpenAlex ID: W2016061239
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description: scientific article; zbMATH DE number 1928898