Results On Uniqueness Of Meromorphic Functions

In 1920s, famous Finland mathematician Rolf Nevanlinna set up the value distribution theory of meromorphic functions, which is one of the greatest theories in mathematics in modern times. The theory is considered to be the basis of modern complex analysis, and it has a very important effect on several complex variables and Diophantine approximation. It also greatly promote the development of modern mathematics.Nevanlinna’s value distribution theory is the main research tool about the uniqueness theory of meromorphic functions. The uniqueness theory of meromorphic functions mainly discuss under what circumstances there is only one function to meet the given conditions. For this kind of problem, many mathematicans have done in-depth research, but R.Nevanlinna’s contributions are the most distinguished. He not only gave many beautiful and concise conclusions, but also the value distribution theory he set up is the main tools of the uniqueness research so far. This paper mainly introduce the applications of value distribution theory on the uniqueness of meromorphic functions.Firstly, for the simple differential polynomial meromorphic functions, Chong-jun Yang and Xinhou Hua(see [12]) gave some concise conclusions in 1997, Mingliang Fang(see [27]) considered share fixed points in 2000; for the u-niqueness problem of differential polynomial integral functions, Ming-liang Fang (see[2],[3],[4]) gave some discussions; Wei-chuan Lin and Hong-xun Yi(see [ll])popularized the related properties to the situation of meromorphic func-tion perfectly, and get the uniqueness of differential polynomial meromorphic function; Junfeng Xu(see [28]) also researched the uniqueness of differential polynomial of meromorphic functions; Xiao-yu Zhang and Jun-fan Chen(see [30])got the uniqueness of meromorphic function in the more general sense.This paper uses the idea of share weight, popularized the conclusion of Junfeng Xu and Hongxun Yi(see [5]) from entire function to meromorphic function, and we obtain several intuitive results.Theorem 1 Let f(z) and g(z) be two nonconstant meromorphic functions, n,m be positive integers, if fn(fm-1)f’and gn(gn-1)g’ share 1 IM and n> 4m+ 16, then f= g.Corollary 1 In theorem 1, let m= 1, then n> 22, we get a more ordinary result:Let f(z) and g(z) be two nonconstant meromorphic functions, n be pos-itive integers, if fn(f-1)f’ and gn(g-1)g’ share 1 IM and n> 22, then f= 9.Theorem 2 Let f(z) and g(z) be two transcendental meromorphic func-tions, n, m be positive integers, if fn(fm-1)f’ and gn(gm -1)g’ share z IM and n>4m+ 18, then f= g.In addition, for the meromorphic functions which have two defective values 0 and oo, Lin Xu and Ying-qiang Yue [27] got three theorems from the view of null point, on the basis of this, this paper take advantage of Nevanlinna three-value theorem and Hong-xun Yi [1] the uniqueness lemma about three functions, and get another type of the theorem and some deductions from the view of the pole.Theorem 3 Let f(z) and g(z) be two nonconstant meromorphic functions, f(z) and g(z) share 1, ∞ CM,if δ(0,f)= δ(0,g)=1 and (?)(∞,f)+(?)(∞,g)>3/2, then, f(z)= g(z) or f(z)g(z)= 1.Corollary 2 Let f(z) and g(z) be two nonconstant meromorphic func-tions, f(z) and g(z) share 0,1,∞ CM, if δ(0, f)= 1,0(∞,f)<3/4, then, f(z)= g(z) or f(z)g(z)= I.Corollary 3 Let f(z) and g(z) be two nonconstant meromorphic func-tions, f{z) and g{z) share 0,1, ∞ CM, if δ(0, f)>1/2, G(∞,f)= 1, then, f(z)= g(z) or f(z)g(z)= l.Corollary 4 Let f(z) and g(z) be two nonconstant meromorphic func-tions, f(z) and g(z) share 0,1, ∞ CM, if δ(0, f)= 1,(?)(∞,f)> 1/2, then, f(z)= g(z) or f(z)g(z)= 1.The dissertation is structured as follows:in chapter 1, we simply intro-duce the basic concepts and primary results of Nevanlinna theory; in chapter 2, we investigate value distribution of one special differential polynomial mero-morphic functions; in chapter 3, we investigate the uniqueness problem of meromorphic function which have defective values, and obtain several better results.