关键词: 整函数, 多复变, 全导数, 微分多项式

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.

Key words: entire function, several complex variables, total derivative, differential polynomial