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Nevanlinna theory of meromorphic functions on annuli. (English) Zbl 1193.30044
In this survey paper, the recent development of Nevanlinna theory of meromorphic functions on annuli is discussed. As the authors mention, this development is motivated historically by its applications to study of growth properties of a solution of complex differential equations, e.g., [Y. M. Chiang and I. Laine, Jap. J. Math., New Ser. 24, No. 2, 367–402 (1998; Zbl 0918.34040)]. Systematic studies on the annuli {z:1/R 0 <|z|<R 0 ,1<R 0 +} are found in e.g., [A. Kondratyuk and I. Laine, Meromorphic functions in multiply connected domains, Fourier series methods in complex analysis. Proceedings of the workshop, 2005. Joensuu: University of Joensuu, Department of Mathematics. Report Series. Department of Mathematics, University of Joensuu 10, 9–111 (2006; Zbl 1144.30013)], in which a number of counterparts of classical theorems in the Nevanlinna theory for meromorphic functions are proved. In Section 2, definitions of Nevanlinna functions in annuli are given. Nevanlinna’s first fundamental theorem in annuli and Nevanlinna’s logarithmic derivatives lemma in annuli are stated in Sections 3 and 4, respectively. The counterparts of other classical theorems, for examples Clunie-type lemmas and the five values theorem are mentioned in Section 5. In Section 6, one of the tools for the proofs so called Valiron’s decomposition lemma is stated. At the end of the paper, the authors state open questions on growth of composite functions and on sequences of annuli.
30D35Distribution of values (one complex variable); Nevanlinna theory
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