亚纯函数正规族的一点注记

研究了一类与Hayman猜想有关的亚纯函数族的正规问题,即函数族中任一函数满足f+a(f(k))n≠b条件下的正规问题,采用顾永兴等(正规族理论及其应用.北京:科学出版社,2007.)的方法讨论了f+a(f(k))n≠b不成立时的正规问题,得到了:设F是区域D内亚纯函数族,k,n(≥k+2)是正整数,a(≠0),b两个有限复常数,若对任意的函数f∈F,f(z)的零点重级至少为k+1,且存在M>0,使得当f+a(f(k))n=b时有|f(z)|≥M,则F在区域D内正规,并对整函数族考虑了分担值时的正规定则的问题.这些结果推广或改进了已有的相关结果.

A Note on Normal Families of Meromorphic Functions

This research is a continuation of the problem of normal families of meromorphic functions about Hayman conjecture,namely some normality criterion under the condition f+a(f(k))n≠ b.By using the method in Gu Yong-xing et al(Normal Family Theory and Its Application.Beijing: Science Press,2007.),we consider the normality criterion without the condition f+a(f(k))n≠b and obtain the following theorem.Let F be a family of meromorphic functions in a domain DC.Let k,n(≥ k+2) be two positive integers and a,b be two non-zero complex constant numbers.Every f∈F has only zeros with multiplicity at least k+1.Suppose there exists a positive constant M>0 such that,for each f∈F,|f(z)|>M when f+a(f(k))n=b,then F is normal in D.Moreover,we consider the normal criterion concerning holomorphic function share values.This result improves or generalizes some previous results.