Uniqueness Of Meromorphic Functions And Some Related Problems Of Normal Families

Value distribution theory of meromorphic functions,was due to R.Nevanlinna in 1920's,and in geometric form by L.Ahlfors about a decade later,is one of the most important achievements in the preceding century to understand the properties of meromorphic functions.The theory is composed of two main theorems,which are called Nevanlinna's first and second theorems that had been significant breakthroughs in the development of the classic function theory,since the Nevanlinna's second theorem generalizes and extends Picard's first theorem greatly,and hence it denoted the beginning of the theory of meromorphic functions.Since then,Nevanlinna theory has been well developed in itself and widely applied to the researches of the uniqueness of meromorphic functions,normal families,complex dynamics and differential equations etc.Meanwhile, in view of the beauty of Nevanlinna theory,many outstanding mathematicians founded and developed the value distribution theory of meromorphic mappings over certain complex manifolds.In 1907,P.Montel introduced the concept of normal family.It plays an important role in complex dynamics.A family of meromorphic functions is called normal if every sequence in the family has a subsequence which converges(locally uniformly with respect to the spherical metric).Our aim is to find the normal families.One guiding principle in the study has been the heuristic principle which says that a family of functions meromorphic(or holomorphic) in a domain and possessing a certain property is likely to be normal if there is no nonconstant function meromorphic(or holomorphic) in the plane which has this property.This is so-called Bloch's principle.Indeed,Bloch's principle is not true in general,but it is still an important guiding principle in the theory of normal family.In 1929,R.Nevanlinna applied the value distribution theory to consider the conditions under which a meromorphic function of a single variable could be determined and derived the famous Nevanlinna's five-value and four-value theorems.From then on,the uniqueness of meromorphic functions of a single variable has been drastically studied by lots of mathematicians and gradually consummated in itself,which recently extended to those of meromophic mappings of several variables.The present thesis involves some results of the author that investigate the normality of some classes of holomorphic functions and the uniqueness of meromophic functions over C and C~m,under the guidance of superwisor professor Hongxun Yi.The dissertation is structured as follows.In Chapter 1,we introduce the general background of Nevanlinna Theory and some notations which are always used in our studies.In Chapter 2,we study the problem of meromrophic functions over C sharing three distinct values and a pair of small functions with weight.Using a lemma of Zhang[60], we prove a result.Indeed,we improve the Nevanlinna's four-value theorem,Brosch's theorem and some results of Alzahary[1,2,3].In Chapter 3,we investigate the uniqueness problem of entire functions that share values or polynomials with their derivatives.By estimating the size ofρ_n's in the famous L.Zalcman's lemma and using the theory of normal family,we deduce this kind of functions have finite order,which is an important property.And then,we obtain some uniqueness theorems,which improve the results given by Rubel and Yang[48]and Li and Yi[40].As an application,we partially solve R.Br(u|¨)ck's conjecture and prove that it holds for the special class of functions F=f~n with n≥2.In Chapter 4,we obtain a result on the normality of holomorphic functions that share a set S with their derivatives,which improves a theorem of Xu[53].Meanwhile, some examples show that the conditions of the result are necessary and the number of elements in the set S is sharp,respectively.Using the theory of normal families, we also prove a uniqueness theorem which was obtained by Li and Yang[40]just with Nevanlinna theory in 1999.In Chapter 5,we deal with the uniqueness problem of meromorphic mappings that share q hyper-planes or moving targets of P~N(C).Meanwhile,some results of Chen and Yan[10,58]are partially improved.