| 钱雪雪,叶亚盛,贾志晶.限制零点个数的亚纯函数的正规定则[J].数学年刊A辑,2019,40(2):165~176 |
| 限制零点个数的亚纯函数的正规定则 |
| Some Normal Criteria of Meromorphic Functions Limited the Numbers of Zeros |
| Received:March 22, 2017 Revised:June 19, 2018 |
| DOI：10.16205/j.cnki.cama.2019.0014 |
| 中文关键词: Meromorphic function, Multiple zeros, Normal family |
| 英文关键词:Meromorphic function, Multiple zeros, Normal family |
| 基金项目: |
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| 中文摘要: |
| 设$k$,$n(\geqslant{k+1})$是两个正整数, $a(\neq0)$,$b$ 是两个有穷复数,
${\cal F}$为区域$\textit{D}$内的一族亚纯函数. 如果对于任意的$f\in\cal{F}$,
$f$的零点重级大于等于${k+1}$, 并且在$D$内满足$f+a[L(f)]^{n}-b$至多有$n-k-1$
个判别的零点, 那么${\cal F}$在$D$内正规.
这里$L(f)=f^{(k)}(z)+a_{1}f^{(k-1)}(z)+\cdots+a_{k-1}f'(z)+a_{k}f(z)$,
其中$a_{1}(z),a_{2}(z),\cdots,a_{k}(z)$是区域$D$上的全纯函数. |
| 英文摘要: |
| Let $k$,$n(\geqslant{k+1})$ be two positive integers, $a(\neq0)$, $b$ be two finite complex numbers and
${\cal F}$ be a family of meromorphic functions in $\textit{D}$. If
for each function $f\in\cal{F}$, all zeros of $f$ have multiplicity
at least $k+1$, and $f+a(L(f))^{n}-b$ has at most $n-k-1$ distinct
zeros in $D$, then ${\cal F}$ is normal in $D$, where
$L(f)=f^{(k)}(z)+a_{1}f^{(k-1)}(z)+\cdots+a_{k-1}f'(z)+a_{k}f(z)$,$a_{1}(z),a_{2}(z),\cdots,a_{k}(z)$
are holomorphic functions in $D$. |
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