复合解析函数族分担亚纯函数的正规性

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.

Nomal families of composite analytic functions and sharing a meromorphic function

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.