Abstract

In this paper, we extend and improve the four-value theorems of Nevanlinna and Fujimoto to the case of meromorphic functions on the annuli. For detail, we will prove that there are at most two admissible meromorphic functions on an annulus sharing a value with multiplicities truncated by two and other three values regardless of multiplicities. We also show that if four admissible meromorphic functions on an annulus share four values regardless of multiplicities then two of them must coincide. Moreover, in our result, the inverse images of these values by the functions with multiplicities more than a certain number do not need to be counted.