摘要:
证明了如下的结论: 设\,$k\geqslant 2$\,是一个正整数, $\mathcal{F}$\,是区域\,$D$\,上的一族全纯函数, 其中每个函数的零点重级至少是\,$k$, $h(z),\,a_1(z),\,a_2(z)\,\cdots,\,a_k(z)$\,是\,$D$\,上的不恒为零的全纯函数. 假设下面的两个条件也成立:\,$\forall f\in\mathcal{F},$ (a) 在\,$f(z)$\,的零点处, $f(z)$\,的微分多项式的模小于\,$h(z)$\,的模; (b) $f(z)$\,的微分多项式不取\,$h(z)$, 则\,$\mathcal{F}$\,在\,$D$\,上正规.
Abstract:
In this paper, we proved: Let $k\geqslant 2$ be a positive integer, $\mathcal{F}$ be a family of holomorphic functions, all of whose zeros have multiplicities at least $k$, and let $h(z)$, $a_1(z)$, $a_2(z)$, $\cdots$, $a_k(z)$ are all nonequivalent to $0$ on $D$. If for any $f\in\mathcal{F}$, the following two conditions are satisfied: (a)~$f(z)=0\Rightarrow |f^{(k)}(z)+a_1(z)f^{(k-1)}(z)+\cdots+a_k(z)f(z)||h(z)|$; (b)~$f^{(k)}(z)+a_1(z)f^{(k-1)}(z)+\cdots+a_k(z)f(z)\neq h(z),$~ where ~$a_1(z), a_2(z),\cdots ,a_k(z)$ and $f$ have no common zeros, then $\mathcal{F}$ is normal on $D$.
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