Normal Families Of Meromorphic Functinons Concerning Differential Polynomial

The theory of normal families of meromorphic functions is an important subject in complex analysis,many scholars who are at home and abroad have made a great number of outstanding contributions to this respect and attained a lof of important results.In this paper we study the normality of families of meromorphic functions concerning differential polynomial.We do a further study in normality of families of meromorphic functions on sharing two values,sharing a value,sharing a set and normality of families of holomorphic functions on sharing functions and concerning exceptional function by using the Nevanlinna's value distribution theory and the theory of normal families.This article is divided into four parts to discuss these issues.In part one,we introduce the research background and the present research situation of the theory of normal families of meromorphic functions and give the main results of this article.In part two,we state some fundamental definitions, usual notations and basic results in Nevanlinna's value distribution theory and the theory of normal families of meromorphic functions.In part three,we discuss families of meromorphic functions on sharing values. In section 3.1 and 3.2,we discuss the normality criterion of families of meromorphic functions concerning sharing two values and sharing a value.We generalize derivatives or high-ordered derivatives of some existing conclusions to differential polynomial situation and weaken the condition to single direction sharing, and obtain theorem 3.1.1,theorem 3.1.2, theorem 3.2.1 and theorem 3.2.2. In part four,we discuss the normality criterion on sharing a set,sharing functions and concerning exceptional function.In section 4.1,we discuss the normality criterion of families of meromorphic functions on sharing a set.We improve derivative of a present result to the case of function multiply its differential polynomial and weaken IM sharing to single direction sharing, and obtain theorem 4.1.1.In section 4.2,we discuss the normality criterion of families of holomorphic functions on sharing functions.We generalize derivatives or high-ordered derivatives of some existing conclusions to differential polynomial situation and replace the constant with holomorphic function,and obtain theorem 4.2.1 and theorem 4.2.2.In section 4.3,we discuss normal families of holomorphic functions concerning exceptional function and improve high-ordered derivative of a present result to the case of function multiply its differential polynomial,and obtain theorem 4.3.1.