涉及微分多项式分担函数的正规定则

Normal Criterion of Shared Function Concerning Differential Polynomials

本文主要讨论了涉及微分多项式分担函数的正规定则,并且得到了以下结果:设F和G为区域D⊂ℂ的两族亚纯函数,所有零点的重级至少为k+1,其中k≥1且为整数。设b(z)≠0在D内全纯,ai,i=1, 2,…,k−1为有穷常数。若G正规,对于G中任意子列{gn},gn⇒g,在区域D上我们有g≢∞和L(g)≢b(z)(其中L(f)=f(k)+ak-1f(k-1)+…+a1f且L(g)=g(k)+ak-1g(k-1)+…+a1g)。若对于任意f∈F,存在g∈G使得:1)f(z)=0⇔g(z)=0;2)f(z)=∞⇔g(z)= ∞;3) L(f(z))=b(z)L(g(z))=b(z);则F在D上正规。

In this paper, we mainly discuss a normal criterion of shared function concerning differential polynomials and proved: Let F and G be two families of functions meromorphic on a domain D⊂ℂ, all of whose zeros have multiplicity at least k+1, where k≥1 is an integer. Let b(z)≠0 be a holomorphic function in the domain D, and ai,i=1, 2,…,k−1 be finite constant. Assume also that G is normal, and for any subsequence {gn},gn⇒g, we have g≢∞ and L(g)≢b(z) on D L(f)=f(k)+ak-1f(k-1)+…+a1f, L(g)=g(k)+ak-1g(k-1)+…+a1g). If for every f∈F, there exist g∈G such that: 1) f(z)=0⇔g(z)=0;2)f(z)=∞⇔g(z)= ∞;3) L(f(z))=b(z)L(g(z))=b(z);Then F is normal on D.