Uniqueness Theorems Of Meromorphic Functions Whose Difference Polynomials Share A Non-zero Polynomial

In1920s, R.Nevanlinna, a famous Finnish mathematician, introduced thecharacterristic functions of meromorphic functions and so founded the valuedistribution theory of meromorphic function. The theory is surely one of the mostimportant achievements in mathematics in the20thcentury because not only it is thebasis of modern meromorphic function theory, but also it has quite effect on thedevelopment of many other mathematical branches, and on the intersection amongthem.Recently，Halburd-Korhonen[16]、Chiang-Feng[25]、Laine and C.C.Yang[19]have founded the Nevanlinna theory in difference and a lot of scholars start theirresearch on the uniqueness of difference of meromorphic functions by using theNevanlinna theory in difference.The present dissertation is the author’ research work under the cordi-al guidance.It consists of three chapters.In chapter one, we briefly introduce some main concepts, usual notati-ons andclassical results with this dissertation in the value distribution theory of meromorphicfunctions.Furthermore，we give the general results of the Nevanlinna theory indifference.In chapter two, we formulate the results on the uniqueness of the difference ofentire functions when sharing a nonzero polynomial CM. And in Chapter3we givethe corresponding theorems considering that the difference polynomial shared thevalue IM.Now we turn to state the main results of the thesis.Theorem1. Let f and g be two distinct transcendental entire functions of finite orders,and let P be a nonzero polynomial.Suppose that η is a nonzero complex number and n≥4is an integer such fhat2deg(P)<n+1. Suppose that fn(z)f(z+η)-P(z)and g"(z)g(z+η)-P(z)share0CM.Then(Ⅰ)If n≥4and ifn(z)f(z+η)/P(z)is a Mobius transformation of g"(z)g(z+η)/P(z),then one of the following two cases will hold:(ⅰ)f=tg,where t≠1is a constant satisfying tn+1=1.(ⅱ)f=eQ and g=te-Q,where P reduces to a nonzero constant c,say, and t≠1is a constant such that tn+1=c2, Q is a nonconstant polynomial.(Ⅱ)If n≥6,then one of the two cases Ⅰ(ⅰ)and Ⅰ(ⅱ)will hold. Theorem2.Suppose that f,g are transcendental meromorphic functions of finite order, α is a nonzero meromorphic functions that ρ(α)<ρ(f), let η be a nonzero complex constant, n and m be two integers that n≥m+6.If fn(z)(fm(z)-1),(z+η)-α(z)and g"(z)(gm(z)-1)g(z+η)-α(z) share0CM,then f=tg,where t is a constant and tm=1.Theorem3Suppose that f,g are transcendental meromorphic functions of finite order and share0CM, η is a nonzero complex constant, let P(z)=anzn+an-1Zn-1+…+a0(an≠0), is a nonzero polynomial, let n be an integer and n>3I-1+2I-2+4.If P(f)f(z+η)and P(g)g(z+η)share1CM,then one of the following cases holds:(1)f=tg,td=1.(2)f=eα,g=ce-α,where α is a polynomial, c is a constant, an2C(n+1)=1.Theorem4Suppose that f,g are transcendental meromorphic functions of finite order and share O CM, η is a nonzero complex constant, let P(z)=αnzn+αn-1zn-1+…+α0(αn≠0), is a nonzero polynomial, n is an integer that degP0<n+1.Suppose that P(f)f(z+η)and P(g)g(z+η)CM share P0(z)CM,if n>2Γ1+1and if F is a Mobius transformation of G or if n>2Γ2+1,then one of the following cases holds:(1)f=tg,td=1.(2)f=eα and g=te-α,where P0degenerated to a constant c, t is a constant and tn+1=c2, α ia a nonconstant polynomial.