Some Normal Criteria of Meromorphic Functions Limited the Numbers of Zeros

DOI：10.16205/j.cnki.cama.2019.0014

 作者 单位 E-mail 钱雪雪 上海理工大学理学院, 上海 200093. 1714774700@qq.com 叶亚盛 上海理工大学理学院, 上海 200093. yashengye@aliyun.com 贾志晶 上海理工大学理学院, 上海 200093. jiazhijing08@163.com

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.

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 $D$.