Survey of Some General Properties of Meromorphic Functions in a Given Domain
The first classical results (principles) related to arbitrary analytic (meromorphic) functions w in a given domain were obtained by Cauchy in 1814–1831, while the next principles had arisen much later, in Ahlfors theory of covering surfaces created in 1935.
In this survey we present some other (diverse type) results of the same generality which were obtained since 1970s.
Previously the most attention was paid to meromorphic functions in the complex plane or in the disks, which were studied in details in the classical Nevanlinna value distribution theory. The results of this survey complement this theory by discovering some new type of phenomena or regularities and, unlike Nevanlinna theory, the result cover all meromorphic functions, including, the most important in application, functions in a given domain.
Some of these results found already applications in other topics, for instance in geometry and complex equations. Meantime the results were presented earlier in some papers devoted to value distribution. Respectively many experts working in other fields are not familiar with them.
The aim of this survey is to facilitate further applications of these results, by presenting them as a collection of some separate formulas or some “ready to use” tools. This will enable to apply these formulas without entering into details and interrelations of these formulas with other topics.
KeywordsMeromorphic functions in a domain Nevanlinna theory Ahlfors theory Gamma-lines Level sets Estimates of derivatives Geometric deficiencies Universal version of value distribution
Mathematics Subject Classification (2010)30A10 30A99 30C99 30D30 30D35 30D99 30F99 30G99
This work was supported by Marie Curie (IIF) award. The author thanks the referee for careful checking.
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