亚纯函数微分多项式IM分担值的唯一性

Uniqueness of differential polynomials of meromorphic functions IM sharing a value

研究了亚纯函数的微分多项式分担一个值的唯一性问题,证明了如果f(z)和g(z)为非常数亚纯函数,其零点和极点的重数至少为s,s为正整数,且满足(n+1)s≥24,n为正整数且n≥2。如果fnf′和gng′分担1 IM,则g(z)=c1ecz,f(z)=c2e-cz,其中c1、c2、c为常数,且满足(c1c2)n+1c2=-1,或者f(z)=tg(z),其中tn+1=1。

This paper study uniqueness of differential polynomials of meromorphic functions sharing 1 IM. Let f(z) and g(z) be two non-constant meromorphic functions, whose zeros and poles are of multiplicities at least s, where s is a positive integer. Let n≥2 be an integer satisfying(n+1)s≥24. If f nf ′ and gng′ share the value 1 IM, then either f(z)=tg(z) for some(n+1)-th root of unity t or g(z)=c1ecz, f(z)=c2e-cz, where c1,c2, c are constants satisfying(c1c2)n+1c2=-1.