Abstract

This paper deals with the finiteness problem of meromorphic funtions on an annulus sharing four values regardless of multiplicity. We prove that if three admissible meromorphic functions $f_1$, $f_2$, $f_3$ on an annulus $\mathbb A({R_0})$ share four distinct values regardless of multiplicity and have the complete identity set of positive counting function, then $f_1=\nobreak f_2$ or $f_2=f_3$ or $f_3=f_1$. This result deduces that there are at most two admissible meromorphic functions on an annulus sharing a value with multiplicity truncated to level $2$ and sharing other three values regardless of multiplicity. This result also implies that there are at most three admissible meromorphic functions on an annulus sharing four values regardless of multiplicities. These results are a generalization and improvement of the previous results on finiteness problem of meromorphic functions on $\mathbb C$ sharing four values.