Survey of some general properties of meromorphic functions in a given domain

5 of April 2018 prof. G. Barsegian from the  Institute of Mathematics of the National Academy of Sciences of Armenia gave a talk dedicated to the 80th anniversary of Prof. Heinrich Begehr at faculty of Mathematics and Mechanics of YSU.

Abstract. 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. These results are interlacing with the classical Nevanlinna value distribution theory studying meromorphic functions in the complex plane or in the disks. The results complement the classical theory by discovering some new type of phenomena or regularities. In addition, they establish analogous phenomena for the most practical in application case when we consider analytic (meromorphic) 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 and its further developments; respectively many experts who could potentially apply these results are not familiar with them. The aim of this survey is to facilitate further applications of these results.

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