• 数学 •

### 多复变整函数涉及全导数的 Picard 型定理

1. 安徽工业大学 数理科学与工程学院, 安徽 马鞍山　243032
• 收稿日期:2020-08-22 出版日期:2021-11-25 发布日期:2021-11-26
• 通讯作者: 杨刘 E-mail:z1721519915@163.com;yangliu6@ahut.edu.cn
• 基金资助:
国家自然科学基金(11701006); 安徽省自然科学基金(1808085QA02)

### Picard-type theorems for entire functions of several complex variables with total derivatives

Shengyao ZHOU(), Liu YANG*()

1. School of Mathematics and Physics, Anhui University of Technology, Maanshan Anhui　243032, China
• Received:2020-08-22 Online:2021-11-25 Published:2021-11-26
• Contact: Liu YANG E-mail:z1721519915@163.com;yangliu6@ahut.edu.cn

Abstract:

In this paper, we use the logarithmic derivative lemma for several complex variables to extend the Milloux inequality to differential polynomials of entire functions. As an application, we subsequently apply the concept to two Picard-type theorems: (1) Let $f$ be an entire function in $\mathbb{C}^{n}$ and $a, b\;(\neq 0)$ be two distinct complex numbers. If $f\neq a, {\cal{P}}\neq b,$ then $f$ is constant. (2) If $f^{s}D^{t_{1}}(f^{s_{1}})\cdots D^{t_{q}}(f^{s_{q}})\neq b$ and $s+$ $\sum_{j = 1}^{q}s_{j}\geqslant 2+\sum_{j = 1}^{q}t_{j},$ then $f$ is constant, where $D^{k}f$ is the $k$ -th total derivative of $f$ and ${\cal{P}}$ is a differential polynomial of $f$ with respect to the total derivative.