The Normality Of Meromorphic Functions Concerning Differential Polynomial And Shared Values

Nevanlinna introduced the definition of characteristic function of meromorphic function in 1925,which attached great importance to the value distribution theory and brought quickly development to value distribution theory.Then in the 1920s,the discovery and application of the first and the second Nevanlinna fundamental theory brought unprecedented development to the value distribution theory.In our country,the research of value distribution theory was started by Xiong Qinglai,who cultivated a research team with other mathematicians majoring in function theory,such as Yang Le,to study the value distribution theory.Based on the research of normality theory,we will discuss the following four parts in this paper:The first part is the introduction.We briefly introduce the development history and status of the value distribution theory.The second part mainly introduces the basic knowledge of the value distribution theory and the normality theory,which include two Nevanlinna fundamental theorems,the definition of characteristic function,logarithmic derivative fundamental lemma,the spherical surface derivative convergence,uniform convergence,the concept of normal,normal criteria of Marty,Zalcman lemma and Hurwitz theorem,etc.The third part is our result and its proof.Embarking from the condition of Hayman problem,we analyze the value distribution theory and normality theory which refer to derivative.The condition that f and g share a value can be extended to the condition that the differential polynomials of f and h share a value,i.e.fn(fn1)(t1)...(fnk)(tk)and k hn(hn1)(t1)...(hnk)(tk)share a value,where nj,tj satisfy(a)nj>tj;(b)n +(?)>3 +(?)tj.The fourth part is the conclusion.We analyze and illustrate the contemporary value distribution theory,and then we conjecture some related normal criteria.