Nevanlinna Value Distribution Theory And Uniqueness Theory Of Meromorphic Functions On Annuli

In this article, we will give some introduction of some important basic knowledges of Nevanlinna value distribution theory and Uniqueness theory in the complex plane and on annuli, for example, The Characteristic Function, Poisson-Jensen’s formula, The First Fundamental Theorem, The Second Fundamental Theorem, Exceptional values and so on, and obtain some important inequalities and a general uniqueness theorem of meromorphic functions on annuli which are also very important in the complex plane. Namely, a generalization of some important inequalities and a general uniqueness theorem of meromorphic functions in the complex plane to on annuli.Ideas as follows:In part one, we introduce why we choose this topic, who is studying now, what we have done and the application.In part two, we give some introduction of important basic knowledges of Nevanlinna value distribution theory and Uniqueness theory in the complex plane and on annuli, for example, The Characteristic Function, Poisson-Jensen’s formula, Exceptional values, The First Fundamental Theorem, The Second Fundamental Theorem, and some related results.In part three, we establish some important inequalities of meromorphic functions on annuli which are also very important in the complex plane, for example, Milloux inequality, Xiong inequality, Hayman inequality, and some other important inequalities. They have a vital role in the middle of many proofs, and more importantly, with these inequalities, more and more theories can be get on the annuli.In part four, we deal with the multiple values and uniqueness of meromorphic functions on annuli. We obtain a general uniqueness theorem of meromorphic functions on annuli, from which an analog of Nevanlinna’s famous five-value theorem is proposed.