On Uniqueness Of Meromorphic Functions Sharing Nonzero Pseudo Common Value And Small Function With Their Derivatives

In 1920s, R.Nevanlinna, a famous Finnish mathematician, introduced the charac-teristic functions of meromorphic functions and so founded two theory,which is called the value distribution theory of meromorphic function. The theory is surely one of the most important achievements in mathematics in the 20th century and is the basis of modern meromorphic functions theory. In about half a century,Nevanlinna theory has been well developed and can be used in the research of the complex differential equations,the uniqueness of meromorphic functions and other fields. In recent decades,the uniqueness of meromorphic functions is more active in international research projects,the study is very rich. the shared values of meromorphic functions start with some study of R. Nevanlinna,who not only laid the basis for the uniqueness of meromorphic functions,but also injected new vitality for the study and development of the uniqueness of meromorphic functions.5IM theory and 4IM theory established by him are classic results in this field. From the end of the twentieth century thirties to the middle of the fifties,the study of the uniqueness of meromorphic functions is stagnated.Until the end of the twentieth century fifties,Qiong Qinglai and Yang Le,the famous mathematician in our country, made some deep results in this respect. With the continuous development and improvement of the uniqueness of meromorphic functions, some problems have been resolved,new research problems are emerging, such as F. Gross question, R.Bruck conjecture, Yi-Yang question and A. Hinkkanen question,are all the subjects of concern to many mathematicians, F. Gross, G. G. Gundersen, G. Frank, E. Mues have gained a lot of research.Professor Hong-Xun Yi did many original work (see e.g.[1][2])on the uniqueness theory of meromorphic functions, and thus drew interest of mathematicians in different countries, which well impelled the development of the uniqueness theory of meromorphic functions and made it more perfect gradually. He has contributed greatly to the promotion of the international position of China in this research area. Professor Xiao-Min Li is more active in the research of uniqueness theory of merom-orphic functions, whose researches are readed by mathematicians from different countries. Furthermore, he also has obtained more achievements in research of complex differential equations and normal families. For example, he has studied the Bruck conjecture and Gundersen question(see e.g.[3][4][5]).The present dissertation is the author' research work under the cordial guidance of Professor. It consists of three chapters.In chapter one, we briefly introduce some main concepts, usual notations and classical results with this dissertation in the value distribution theory of meromorphic functions.In chapter two,we study uniqueness problem for meromorphic functions whose derivatives share four small functions or values,and obtain two main uniqueness theorems which are provements of results given by R.Nevanlinna, G.G.Gundersen,L.Yang,G.D.Qiu and other authors.Now we turn to state the main results of the thesis.Theorem 1 Let f and g be two nonconstant meromorphic functions,and let b1,b2,b3,b4 be four distinct elements in{S(f)∩S(g)}.If (?)1)(0,f(k)-bj)= (?)1)(0,g(k)-bj) (j=1,2,3,4),wherel(≧1) and k(≧1)are positive integers such that 1/l+1/k+1·(1+1/l)<1,then f(k)=g(k). Theorem2 Let f and g be two nonconstant meromorphic functions, let b1,b2,b3 be three distinct finite values.If f(k) and g(k) are two distinct nonconstant meromorphic functions that share b1,b2,b3,∞IM,where k(≧1) is a positive integer,then f and g are functions with normal growth,f and g have same order of growth,and the order of f and g is a positive integer or infinite.In chapter three,we study uniqueness problem for meromorphic functions whose nonlinear differential polynomials have one nonzero pseudo common value, and obtain a main uniqueness theorem which improves the corresponding ones given by C.Y Fang and M.L.Fang., I.Lahiri and Mandal, A.Banerjee,and others.Theorem 3 Let f and g be two nonconstant meromorphic functions such that E2)(1,fn(af2+bf+c)f')=E2)(1,gn(ag2+bg+c)g'),where a≠0,b,c are complex numbers satisfying |b|+|c|≠0,n is a positive integer such that n>16-7 max{(?)(∞,f),(?)(∞,g)}. Then one of the following four cases holds:(i)If b≠0,c=0 and(?)(∞,f)+(?)(∞,g)>4/n+2,then f=g.(ii) If b≠0 and c≠0 such that az2+bz+c=0 has two distinct roots,and if one of f and g is not an entire function whose poles is of multiplicities≥2, then f=g.(iii)If b≠0 and c≠0 such that az2+bz+c=0 has two equal roots, then f=g.(iv)If b=0 and c≠0,then f=g or f=-g. If n is one even number, then f=-g can not occur.