The Normality Of Meromorphic Functions Involving Differential Polynomial

After P.Montel introducing the definition of the normal family, the value distribu-tion theory got new development in 1970 and it has the extremely vital significance:then in the 1920s,the introduction of R.Nevanlinna the first and the second fundamental theo-rem have brought the unprecedented development of value distribution theory;the found of Marty formal rule and Zalcman lemma brought the theory basis to reversely think-ing normality and gave new ideas and vision for researchers to find regularity.After then,some Chinese mathematicians, especially including Xiong Qinglai,Yang Le,Gu Yongxing and Fang Mingliang.They have made a important contribution to the devel-opment of the value distribution theory, what’s more,their study is at an international level. So based on the theory of normality, discussion in this paper mainly divided into five parts:The first part is the introduction, analysis and the status of the development of the value distribution theory.The second part mainly introduces the basic knowledge of value distribution the-ory (Nevanlinna first and the second fundamental theorem, the characteristic function, logarithmic derivative fundamental lemma, etc.) and the basic knowledge of the the-ory of normality (according to the spherical surface derivative convergence, uniform convergence, the concept of formal, Marty formal rule), Zalcman lemma and Hurwitz theorem.The third part embarks from the basic problem of Hayman, condition ’f≠0’ ’f(n)≠1’, Weakened into a ’fnf’≠ 1’, and then ’fn-f’≠1’, And conditions ’n≥ 4’. Also gradually become weak to ’n≥2’, Besides, gradually push the exceptional values to the exceptional function and function column and improve the conditions.The fourth part started by Hayman problem,but different from the second part.It considers the aspect of Shared values, which is similar to the second part in the aspect of gradually weakening condition, making the resulted theorem more practical.The fifth part is the conclusion, prospect to the contemporary value distribution theory and making assumptions and speculation to the corresponding formal rule.