正规族与分担值

Shared Values and Normal Families

文章证明了若F为区域D内一族亚纯函数,其每个函数的零点重数至少为k+l,l≥2,k为正整数,若对f∈F有:f(k)(z)=0f(k+l)(z)=0,f(k+l)(z)=b→f(k)(z)=b则F在区域D内正规,其中b≠0.若F={f(z)}为区域D内一族亚纯函数,F(k)(z)={f(k)(z)|f(z)∈F},若对f(z)∈F有f(z)的零点重数至少为k,且存在常数A≥1,使得f(z)=0→|f(k)(z)|≤A,则当F(k)在D上正规时,F也正规.

This paper proves the following theorem：Let F be a family of meromorphic functions in a domain D,k and l≥2 be positive integers,and b a non-zero complex number. If for each f∈F,the zeros of f（z）are of multiplicity at least k＋l,and f（k）（z）=0f（k＋l）（z）=0,f（k＋l）（z）=b→f（k）（z）=b,then F is normal in D. any family F of meromorphic functions in a domain, whose every member＇s zeros are of multiplicity at least k and the member＇s k-order derivative takes values at the zeros with a uniform finite boundary,will be normal if the family consisted of the k-order derivatives of all member＇s of F is normal.