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Nevanlinna theory in an annulus. (English) Zbl 1102.30025
Barsegian, G. (ed.) et al., Value distribution theory and related topics. Boston, MA: Kluwer Academic Publishers (ISBN 1-4020-7950-8/hbk; 1-4020-7951-6/e-book). Advances in Complex Analysis and its Applications 3, 167-179 (2004).
Presentation of Nevanlinna theory in $$| z| \geq R$$, i.e. outside of a disc, has been considered in the form of brief remarks only in several articles, see e.g. [L. Bieberbach, Theorie der gewöhnlichen Differentialgleichungen auf funktionentheoretischer Grundlage dargestellt. 2. umgearb. und erweit. Aufl. Die Grundlehren der mathematischen Wissenschaften. 66. (Berlin-Göttingen-Heidelberg-New York): Springer-Verlag. (1965; Zbl 0124.04603)] and S. Bank and I. Laine [J. Reine Angew. Math. 344, 1–21 (1983; Zbl 0524.34007)]. The purpose of the present paper is to collect these remarks into a complete presentation, generalizing them for functions meromorphic in an open annulus. In particular, the paper contains counterparts of the first and second main theorem as well as of the lemma of the logarithmic derivative. As an example of further considerations of the theory, an annular version of the standard Clunie lemma will be given.
For the entire collection see [Zbl 1081.30001].

##### MSC:
 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory