亚纯函数的正规定则

Normal Criterion of Meromorphic Functions

文章通过幅角原理和Zalcman-Pang引理,利用构造辅助函数的方法,对满足条件f′-a(z)f k≠b(z)的亚纯函数族的正规性进行研究。首先,讨论当k=3,例外函数a(z),b(z)均为全纯函数时,在函数族零点重级的条件下,亚纯函数族是正规的,完善了正规族已有结论;其次,将例外函数由全纯函数推广为亚纯函数,讨论例外函数a(z),b(z)分别为亚纯函数时的正规性,得到当a(z)是亚纯函数时,全纯函数族是正规的,当b(z)是亚纯函数时,无零点的亚纯函数族是正规的,在一定范围内推广了亚纯函数的现有结果。

This paper studies the regularity of the meromorphic function family satisfying the condition f′−a(z)f k≠b(z)through the amplitude angle principle and the P-Z lemma,and the method of constructing auxil⁃iary functions.First of all,it is discussed that when k=3,omitted function a(z),b(z)are holomorphic functions,un⁃der the condition of zero weight of the function family,the meromorphic function family is normal,and the normal family has been concluded.Secondly,we generalize the omitted function from a holomorphic function to a meromor⁃phic function,discuss as the regularity of the omitted function a(z)and b(z)when it is a meromorphic function,and obtain that when a(z)is a meromorphic function,the family of holomorphic functions is normal.And when b(z)is a meromorphic function,the family of non-zero meromorphic functions is normal,and generalizes the existing results of meromorphic functions from a certain range.