亚纯函数分担有理函数的唯一性

Uniqueness of Meromorphic Functions Sharing a Rational Function

采用权弱分担值的思想讨论两个亚纯函数fnf′,gng′权弱分担有理函数的唯一性,得到:设p(z),q(z)为两个互质的n1,n2次多项式,f,g为两个非常数超越亚纯函数,如果fnf′与gng′分担"(pq((zz)),m)"且(1)当2≤m≤∞时,满足n≥max{11,2n1+4n2+3};(2)当m=1时,满足n≥max{13,2n1+4n2+3};(3)当m=0时,满足n≥max{23,2n1+4n2+3},则f=c1Q(z)exp(α(z)),g=c2Q-1(z)exp(-α(z)),其中:c1,c2为2个常数且Q(z)是有理函数;α(z)为满足(c1c2)n+1(Q′(z)/Q(z)+α′(z))2≡(p(z)/q(z))2的多项式,或者f=tg,t为常数且满足tn+1=1.

In this paper,we study the uniqueness of two meromorphic functions fnf′,gng′ weakly weighted sharing a rational function and the following result is proved：let p（z）,q（z） be two co-prime polynomials of n1,n2 respectively,let f,g be two non-constant transcendental meromorphic functions.If fnf′ and gng′ share ＂（p（z）q（z）,m）＂ and（i） when 2≤m≤∞;n≥max{11,2n1＋4n2＋3};（ii） when m=1,n≥max{13,2n1＋4n2＋3};（iii） when m=0,n≥max{23,2n1＋4n2＋3},then f=c1Q（z）exp（α（z））,g=c2Q-1（z）exp（-α（z））,where c1,c2 are two constants,Q（z） is a rational function,and α（z） is a non-constant polynomial satisfying（c1c2）n＋1（Q′（z）/Q（z）＋α′（z））2≡（p（z）/q（z））2 or f=tg for a constant t satisfying tn＋1=1.