The partially shared values and small functions for meromorphic functions in a k-punctured complex plane

The main aim of this article is to discuss the uniqueness of meromorphic functions partially sharing some values and small functions in a k-punctured complex plane Ω. We proved the following: Let f1,f2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f_{1},f_{2}$\end{document} be two admissible meromorphic functions in Ω and αj(j=1,2,…,l)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha _{j}\ (j=1,2,\ldots ,l)$\end{document} be l(≥5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$l(\geq 5)$\end{document} distinct small functions with respect to f and g. If E˜(αj,Ω,f1)⊆E˜(αj,Ω,f2)(j=1,2,…,l)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\widetilde{E}(\alpha _{j},\varOmega ,f_{1})\subseteq \widetilde{E}(\alpha _{j},\varOmega , f_{2})\ (j=1,2,\ldots ,l)$\end{document} and lim infr→+∞∑j=1lN‾0(r,1f1−αj)∑j=1lN‾0(r,1f2−αj)>52l−5,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \liminf_{r\rightarrow +\infty }\frac{\sum_{j=1}^{l}\overline{N} _{0} (r,\frac{1}{f_{1}-\alpha _{j}} )}{\sum_{j=1} ^{l}\overline{N}_{0} (r,\frac{1}{f_{2}-\alpha _{j}} )}> \frac{5}{2l-5}, $$\end{document} then f1≡f2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f_{1}\equiv f_{2}$\end{document}. Our results are some improvements and extension of previous theorems given by Cao–Yi and Ge–Wu.