Uniqueness Of Meromorphic Functions Concerning The Difference Operator And Shift Operator

The research of the function value distribution theory based on Nevalinna theory is an important research direction in the field of complex analysis. In this paper,we studies the uniqueness problem of meromorphic function which difference operator,shift operator and difference polynomials sharing common values. Generalizing some recent results,we obtain some following main results:Theorem1Let f(z),g(z)be transcendental meromorphic functions with finite order. Suppose that n(≥14),k(≥3)are positive integers and c is a nonzero complex constant.If Ek(1,fnf(z+c))=Ek(1,gng(z+c)),then we get:f(z)≡t1g(z) or f(z)g(z)≡t2,where t1,t2is a constant satisfying t1n+1=1,t2n+1=1.Theorem2Let f(z),g(z) be transcendental entire functions with finite order. Suppose that n(≥17) is a positive integer and c is a nonzero complex constant.If Ek(1,fn△cf)=Ek(1,gn△cg),then:f(z)≡t1g(z) or f(z)g(z)≡t2,where t1,t2is a constant satisfying t1n+1=1,t2n+1=1.Theorem3Let f(z),9(z) be nonconstant meromorphic functions with finite order and c is a nozero complex constant.f(z)=∞(?)g(z)=∞, f(z+c)=1(?)g(z+c)=1.If N2(r,1/f)+N2(r,1/g)+2N(r,f)<(λ+o(1))T(r), where λ<1,T(r)=max{T(r,f),T(r,g)}.Then f(z)≡g(z) or f(z)g(z)≡1.We will talk about the product of the function’s shift operator in the next. Theorem4Let f(z),g(z) be transcendental entire functions with finite order.aj,bj,cj(j=1,2,...,s)are complex constant.F(z)=∏j=1sajf(z+cj),G(z)=∏j=1sbjg(z+cj).If f(z)=0(?)g(z)=0,F(z)=1(?)G(z)=1, and δ(0,f)>2/3,then we have F(z)≡G(z) or F(z)G(z)≡1.Theorem5Let f(z),g(z) be transcendental entire functions with finite order.α(z) is a small function of f(z),g(z),cj(j=1,2,…,s) are different complex constants,m,n,s,μj(j=1,2,...,s)are non-negative integers,σ=∑j=1sμj.If(fn(fm-1)∏j=1sf(z+cj)μj)(k)=α(z)(?)(gn(gm-1)∏j=1sg(z+cj)μj)(k)=α(z),and n≥5k+4m+4σ+8,We have f(z)=tg(z),where tm=1.Considering the meromorphic functions,we get the following results.Theorem6Let f(z),g(z) be nonconstant meromorphic functions satisfy-ing ρ(f)<∞,ρ(g)<∞.f(z),g(z) share∞IM.α(z)(?)0is an entire function satisfying ρ(α)<ρ(f).m,n,s,μj(j=1,2,...,s)are non-negative integers,σ=∑j=1sμj.cj(j=1,2,…,s)are nonzero complex constants.F(z)=fn(fm1)∏j=1sf(z+cj)μj,G(z)=gn(gm-1)∏j=1sg(z+cj)μj.F(z)-α(z),G(z)-α(z) share0,∞CM.If n≥m+2s+3σ+7we get f(z)=tg(z),where tm=1.