华东师范大学学报(自然科学版) ›› 2012, Vol. 2012 ›› Issue (3): 61-70.

• 应用数学与基础数学 • 上一篇    下一篇

涉及微分多项式及例外函数的正规定则

王 雪1, 刘晓俊2, 陈巧玉3   

  1. 1. 阜阳师范学院~~数学系, 安徽~阜阳 236041; 2. 上海理工大学~~数学系, 上海 200093; 3. 华东师范大学~~数学系, 上海 200241
  • 收稿日期:2011-06-10 修回日期:2011-09-01 出版日期:2012-05-25 发布日期:2012-05-22

Normal criterion concerning differential polynomials and omitted functions

WANG Xue 1, LIU Xiao-jun 2, CHEN Qiao-yu 3   

  1. 1. Department of Mathematics, Fuyang Normal College, Fuyang Anhui 236041, China; 2. Department of Mathematics,University of Shanghai for Science and Technology, Shanghai 200093, China; 3. ,Department of Mathematics, East China Normal University, Shanghai 200241, China
  • Received:2011-06-10 Revised:2011-09-01 Online:2012-05-25 Published:2012-05-22

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.

关键词: 全纯函数, 微分多项式, 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 $D$.

Key words: holomorphic function, differential polynomial, normal

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