A Uniqueness Theorem for Meromorphic Functions Concerning Total Derivatives in Several Complex Variables

Motivated by many differences for the total derivative between entire functions and meromorphic functions, we mainly investigate the uniqueness problems for meromorphic functions in several complex variables concerning the total derivatives. Let f and g be two nonconstant meromorphic functions on Cm,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}^{m},$$\end{document}k be a positive integer such that f=0⇔g=0,Dkf=∞⇔Dkg=∞,Dkf=1⇔Dkg=1.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f=0\Leftrightarrow g = 0, D^{k}f=\infty \Leftrightarrow D^{k}g=\infty , D^{k}f=1\Leftrightarrow D^{k}g=1.$$\end{document} We get that if 2δ(0,f)+(k+4)Θ(∞,f)>k+5,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\delta (0, f)+(k+4)\varTheta (\infty , f)>k+5,$$\end{document} then Dkf-1Dkg-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{D^{k}f-1}{D^{k}g-1}$$\end{document} is a nonzero constant. This is an extension of a uniqueness theorem for entire functions due to L. Jin. There are several examples to show that our result is sharp.