涉及微分多项式分担函数的正规定则

Normal Criterion of Shared Function Concerning Differential Polynomials

摘要: 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，a_i,i=0, 1,…,k−1为有穷常数。若G正规，对于G中任意子列{g_n}，g_n⇒g，在区域D上我们有g≢∞和L(g)≢b(z)(其中L(f)=f^(k)+a_k-1f^(k-1)+…+a₀f且L(g)=g^(k)+a_k-1g^(k-1)+…+a₀g)。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。

Abstract: 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 a_i,i=0, 1,…,k−1 be finite constant. Assume also that G is normal, and for any subsequence {g_n},g_n⇒g, we have g≢∞ and L(g)≢b(z) on D L(f)=f^(k)+a_k-1f^(k-1)+…+a₀f, L(g)=g^(k)+a_k-1g^(k-1)+…+a₀g). 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.