| 谭卫平,詹小平.亚纯函数微分多项式f^kQ[f]+P[f]的零点分布[J].数学年刊A辑,2013,34(3):365~372 |
| 亚纯函数微分多项式f^kQ[f]+P[f]的零点分布 |
| The Distribution of Zeros for Differential Polynomials $f^kQ[f]+P[f]$ of Meromorphic Function |
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| DOI： |
| 中文关键词: 整函数, 亚纯函数, 例外集, 微分多项式, ε集 |
| 英文关键词:Entire function, Meromorphic function, Exceptional set,
Differential polynomial, εset |
| 基金项目:湖南省教育厅科研基金(No.09C228) |
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| 中文摘要: |
| 研究了超越亚纯函数$f$的微分多项式$f^kQ[f]+P[f]$的零点分布.
给出了以下结果:对于满足$\delta(\infty,f)\geq1-\alpha>0$ ($\alpha$为常数, $0\leq \alpha<1$ )的超越亚纯函数$f(z)$,
若$T(r,f)=O((\log r)^2)$,则微分多项式$f^kQ[f]+P[f]$ ($Q[f]\not\equiv 0,\ P[f] \not\equiv 0$)在
可数个圆盘并集之外有无穷多个零点,其中$k>\frac{1+\Gamma_{P}+\gamma_{P}+\alpha(1+\Gamma_Q+\Gamma_{P}-\gamma_{P})}
{1-\alpha }$, $\Gamma_{Q}$是$Q[f]$的权, $\Gamma_{P}$和$\gamma_{P}$是$P[f]$的权和次数. |
| 英文摘要: |
| This paper discusses the distribution of zeros for differential polynomial $f^kQ[f]+P[f]$ of trancendental meromorphic function $f$, and gives following result:
Letting $f$ be a trancendental meromorphic function with $\delta(\infty,f)\geq1-\alpha>0\ (\alpha$ is a constant, $0\leq\alpha<1)$, such that $T(r,f)=O((\log r)^2)$,
then the differential polynomial $f^kQ[f]+P[f]\ (Q[f]\not\equiv 0,P[f]\not\equiv 0)$ has infinitely many zeros outside the union of infinitely many discs,
where $k>\frac{1+\Gamma_{P}+\gamma_{P}+\alpha(1+\Gamma_Q+\Gamma_{P}-\gamma_{P})}{1-\alpha }$, $\Gamma_Q$ is the weight of $Q[f]$,
$\Gamma_P$ and $\gamma_P$ are the weight and the degree of $P[f]$, respectively. |
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