摘要
本文推广了Bergweiler的一个正规定则:设α(z)和F分别是区域D上的非常数解析函数与解析函数族,R(z)是一个次数不低于2的有理函数.如果对族F中函数f(z)和g(z),Rof(z)和Rog(z)分担α(z)IM,并且下述条件之一成立:(1)对任意z0∈D,R(z)-α(z0)有至少两个不同的零点或极点;(2)存在z0∈D使得R(z)-α(z0):=P(z)Q(z)仅有一个零点(或极点)β0,同时k=lp(或k=lq),其中l和k分别是f(z)-β0和α(z)-α(z0)在z0处的零点重数,P(z)和Q(z)分别是次数为p和q的互质的多项式,并且α(z0)∈C∪{∞}.那么F在D内正规.
A result of Bergweiler is extended and an alternative proof is given in this paper.Our main result is as follows:Letα(z)be a nonconstant meromorphic function,F be a family of analytic functions in a domain D,and R(z)be a rational function of degree at least 2.If Rof(z)and Rog(z)shareα(z)IM for each pair f(z),g(z)∈F and one of the following conditions holds:
(1)R(z)-α(z0)has at least two distinct zeros or poles for any z0∈D;
(2)There exists z0∈D such that R(z)-α(z0):= P(z) /Q(z) has only one distinct zero(or pole)β0 and suppose that the multiplicities l and k of zeros of f(z)-β0 andα(z)-α(z0)at z0,respectively,satisfy k≠lp(or k≠lq),possibly outside finite f(z)∈F,where P(z)and Q(z)are two of no common zero polynomials with degree p and q respectively,andα(z0)∈C∪{∞}.
Then F is normal in D.
Some examples are given to illustrate that the conditions in above result are necessary.
出处
《中国科学:数学》
CSCD
北大核心
2010年第5期429-436,共8页
Scientia Sinica:Mathematica
基金
国家自然科学基金(批准号:10771220)
教育部博士点基金(批准号:200810780002)资助项目
关键词
复合函数
正规定则
分担值
meromorphic function
normal family
rational function
share value