Shared Values And Normality Criteria Of Meromorphic Functions

For decades, meromorphic function theory keeps developing and obtains series of systematic and profound results under the influence of Nevanlinna theory, especially the development of the uniqueness theory, value distribution theory and normal family theory.E.Muse,F.Gross,W.Schwick,G.Gundersen,W.Bergweiler, the foreign complex analysis experts , and the domestic such as Q.L.Xiong,Q.T.Zhang,L.Yang,H.X.Yi,Y.X.Gu,Y.F.Wang,L.Z.Yang,X.C.Pang, all of who have made lots of remarkable achievements in the above ares. These results not only promote the development of the theory of meromorphic functions, but also have quite an effect on the related mathematic branches, for instance, complex dynamical systems, complex differential equations, meromorphic functions on P-adic and non-Archimedeam fields, and the research of meromorphic map on high space even on general manifolds. Currently, with the intercross among related disciplines and research areas, which all base on the theory of meromorphic functions, new research methods have generated. On the one hand, some nodes have been broken through. On the other hand, new research issues are gradually emerging. For example, many important questions on shared values and value distribution which need urgently dealt with.Because Nevanlinna theory is the basement of normal families, so it has a sufficient basis of using normal family theory to study share values and value distribution of meromorphic functions.Under the cordial guidance of professor Jian-Ping Wang, the present dissertation is part of the author's research work on normal families problems of functions in uniqueness theory of meromorphic functions, it consists of three parts.In chapter 1, we mainly introduced some fundamental concepts, important results and common notations related to shared values and normal families of meromorphic functions, the development and present situation on the theory of normal families, the purpose of this topic.In chapter 2, we studied the normality of meromorphoc functions, and proved two normal criteria of meromorphic functions with multiple zeros. Combinating of differential polynomial, we proved Theorem 1. From the perspective of sharing one value, we proved Theorem 2. The results are the following: Theorem 1 Let k (≥2) be a positive integer, let D be a domain in the complex plane C , are holomorphic functions in D and , , let F be a family of meromorphic functions in D and satisfy that (i) ?f∈F, the multiple of the poles of f at least 2; (ii) for 2≤k≤4, ?f∈F,the zeros of f have multiplicity at least k + 1; (iii) for k≥5,,the zeros of f have at least k . Define If there exists a constant M > 0 such that for any f∈F, and any z∈E ( f),we have then F is normal in D .Corollary 1 Let k (≥2) be a positive integer, let D be a domain in the complex plane C , let h( z ) be a holomorphic function in D ,let F be a family of meromorphic functions in D and satisfy that for 2≤k≤4, ?f∈F,the zeros of f have multiplicity at least k + 1;for k≥5, ,the zeros of f have multiplicity at least k .If for any f∈Fand any z∈E ( f),we have .Corollary 2 Let k (≥2) be a positive integer, let D be a domain in the complex plane C , let are holomorphic functions in D and , let F be a families of meromorphic functions in D and satisfy that (i), the multiple of the poles of f at least 2; (ii) for 2≤k≤4, ,the zeros of f have multiplicity at least k + 1; (iii) for k≥5, ,the zeros of f have at least k .Define If there exists a constant M > 0 such that for any f∈F , f≠0 and any z∈E ( f),we have f ( k) ( z )≤M, then F is normal in D .Corollary 3 Let k (≥2) be a positive integer, let D be a domain in the complex plane C , let , L , ak ( z) are holomorphic functions in D and, let F be a family of meromorphic functions in D and satisfy that (i) , the multiple of the poles of f at least 2; (ii) for 2≤k≤4, ?f∈F,the zeros of f have multiplicity at least k + 1; (iii) for k≥5, ,the zeros of f have at least k .If ,we have then F is normal in D .Corollary 4 Let k (≥2) be a positive integer, let D be a domain in the complex plane C , let , L , ak ( z) are holomorphic functions in D and , let F be a family of meromorphic functions in D and satisfy that (i) ?f∈F, the multiple of the poles of f at least 2; (ii) for 2≤k≤4, ,the zeros of f have multiplicity at least k + 1; (iii) for k≥5, ,the zeros of f have at least k .If ,we have then F is normal in D .Theorem 2 Let k (≥2) be a positive integer, letαbe a nonzero finite complex number, let F be a family of meromorphic functions in the plane domain D and satisfy that (i) , the poles of f have multiplity at least 2;(ii) for 2≤k≤4, , the zeros of f have multiplicity at least k + 1; for k≥5, , the zeros of f have multiplicity at least k . If for each pair and shareαIM, then F is normal in D .