分担值与正规族

设F是平面区域D上的亚纯函数族,a,b是两个有穷非零复数.如果■ff∈F,f(z)=a■f~((k))(z)=a,ff~((k))(z)=b■f~((k+1))(z)=b,且f-a的零点重数至少为k(k≥3),那么函数族F在D内正规;当k=2时,在条件a≠4b的情况下,同样有函数族F在D内正规.

Sharing Values and Normal Families

Let $\mathcal{F}$ be a family of meromorphic functions on domain $\mathbf{D}$, $a, b$ be two non-zero distinct finite complex numbers. If $\forall f \in \mathcal{F}$, $f(z)=a \Rightarrow f^{(k)}(z)=a$, $f^{(k)}(z)=b \Rightarrow f^{(k+1)}(z)=b$, and all zero points of $f-a$ are of multiplicity at least $k \ (k \geq 3)$, then $\mathcal{F}$ is normal on $\mathbf{D}$; in the case of $k=2$, if $a \neq 4b$, then $\mathcal{F}$ is also normal on $\mathbf{D}$.