Journal of East China Normal University(Natural Science) ›› 2021, Vol. 2021 ›› Issue (6): 38-46.doi: 10.3969/j.issn.1000-5641.2021.06.005

• Mathematics • Previous Articles     Next Articles

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;


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

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