The Problem Of Value Sharing Of Meromorphic Of Entire Functions

Sharing problem of meromorphic functions and entire functions is an im-portant subject in complex analysis.In 1920',Nevanlinna established one of the most important theories in that century, which is value distribution theory of meromorphic functions. The theory is now called Nevanlinna theory in honor of him. From then on,in nearly one hundred years,with the development of this theory,the uniqueness of meromorphic functions,normal families and com-plex differential equations etc. have been developed by many mathematicians, for example,E.Muse,G,Frank,N.Steinmetz,F.Gross,I.Laine and Q.L.Xiong, L.Yang.And they have made a large number of outstanding contributions and developed a number of important results.This paper involves some results of the author under the guidance of profes-sor HongXun Yi,which is the problem of value sharing of meromorphic functions or entire functions.The structure of the paper is aS follows.In Chapter 1,we can see some basic knowledge of Nevanlinna Theory.In Chapter 2,we study the problem of uniqueness of sharing one value of meromorphic functions,mainly improve the theorem 1 in Xiao-Yu Zhang and Jnn—Fan Chen[14]and get theorem 2.1:Theorem 2.1. Let f and g be two nonconstant meromorphic functions and share∞IM；Let n,k and m be three positive integers with n>3m+ 7k+11,and let P(z)=amZm+am-1zm-1+…+a1z+α0 or P(z)三c0, whereα0≠0,αl,...,am-1,αm≠0,c0≠0 are complex constants. LetF= [fn(z)P(f)](k),G=[gn(z)P(g))](k),and E1)(1,F)=E1)(1,G)；then(i)when P(z)=amzm+am-1Zm-1+...+α1z+α0,either f(z)≡tg(z)for a constant t Such that td=1,where d=(n+m,...,n+m-i,…,n),am-i≠0 for some i=0,1,...,m,orf(z)和g(z)satisfly R(f,g)三0,where R(ω1,ω2)＝ω1n(αmω1m+am-1ω1m-1+...+α0)-ω2n(αmω2m+am-lω2m-1+...+α0)； (ii)when P(z)三c0,either,(z)=c1/(?),g(z)=c2/(?),where c1,c2 and c are three constants satisfying(-1)k(c1c2)n(nc)2k=1,or f(z)=t9(z),for a constant t such that tn=1.In Chapter 2,we consider the problem of uniqueness of entire function shar-ing a small function with its differential polynomials,mainly improve the theorem 1.1 and theorem 1.2 in Jun Wang[16] and get theorem 3.1 and theorem 3.2:Theorem 3.1.Let f be an entire function of finite order,σ(f)≠1,αis a small function of f(z).Let L(f)=αkf(k)αk-1f(k-1)+…+α1f'+α0 f is a differential polynomial of f(z),whereαk(≠0),ak-1,….,α1,α0 are constants.If f与L(f)shareαCM,then L(f)-α=c(f-α)for some non-zero constant c.Theorem 3.2.Let f be an entire function of finite order,σ(f)≠1,αis a small fulnction of,(z).Let L(f)=αkf(k)+αk-1f(k-1)+…+α1f'+α0f is a differential polynomial of f(z),whereαk(≠0),ak-1,...,α1,α0 are constants.If f与L(f)share a IM,and then L(f)-α=h(z)(f-α) where h(z)is a meromorphic function of order no more than s.