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

Abstract

Abstract 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 $f_{1},f_{2}$f1,f2 be two admissible meromorphic functions in Ω and $\alpha _{j}\ (j=1,2,\ldots ,l)$αj(j=1,2,…,l) be $l(\geq 5)$l(≥5) distinct small functions with respect to f and g. If $\widetilde{E}(\alpha _{j},\varOmega ,f_{1})\subseteq \widetilde{E}(\alpha _{j},\varOmega , f_{2})\ (j=1,2,\ldots ,l)$E˜(αj,Ω,f1)⊆E˜(αj,Ω,f2)(j=1,2,…,l) and $$ \liminf_{r\rightarrow +\infty }\frac{\sum_{j=1}^{l}\overline{N} _{0} (r,\frac{1}{f_{1}-\alpha _{j}} )}{\sum_{j=1} ^{l}\overline{N}_{0} (r,\frac{1}{f_{2}-\alpha _{j}} )}> \frac{5}{2l-5}, $$lim infr→+∞∑j=1lN‾0(r,1f1−αj)∑j=1lN‾0(r,1f2−αj)>52l−5, then $f_{1}\equiv f_{2}$f1≡f2. Our results are some improvements and extension of previous theorems given by Cao–Yi and Ge–Wu.