The Generalization Of The Mues Lemma On Small Functions And Its Use In Nevenlinna Theorem

The uniqueness theory of meromorphic functions mainly studies conditions underwhich there is only one function or if we can obtain a group of functions such thatsatisfying some relationships and also satisfying the given conditions.For this new subject, the value distribution theory founded by R.Nevanlinna in1920's has become the main tool. In recent decades, many mathematicians have con-tributed to the research of this theory. A lot of deep results were obtained by XiongQinlai and Yang Le. Many other mathematicians such as E.Mues, G.Gunderson andM.Reinders also obtained a lot of elegant results on the research of the uniquenesstheory. Mues Lemma plays a great role in the uniqueness theorem of meromorphicfunctions, and many results have been obtained based on this lemma. However, with-out generalizing the Mues Lemma to small functions, it is not possible to achieve moreresults in the research of small functions.In this paper, I present a generalization of the Mues Lemma on small functions andproved it, thus, totally solve this problem which bothered many peoples, also hope thatit may help motivate research in this direction. Afterward, a prove of the generalizationof two lemmas in small functions and an example of 4DM theorem will be given usingthe generalization of Mues lemma in small function.In this paper, I mainly introduced the work I had done under my teacher YaoWeihong's help. The paper is diveded into 6 sections.