On the characteristics of meromorphic functions that share three values CM (Q1269637)

Let \(f\) and \(g\) be two functions meromorphic in the complex plane and \(a\) be a value in the extended complex plane. It is said that \(f\) and \(g\) share the value \(a\) CM provided that \(f\) and \(g\) have the same \(a\)-points counting multiplicities. In this paper, by constructing an auxiliary function and analyzing the counting function of common 1-points of \(f\) and \(g\), the authors obtained a more precise relation between the Nevanlinna characteristics \(T(r,f)\) and \(T(r,g)\) when \(f\) and \(g\) share three values CM. As applications, some known results can be improved. The main theorem is the following: Let \(f\) and \(g\) be two non-constant meromorphic functions sharing 0, 1 and \(\infty\) CM. Then for any positive number \(\varepsilon\), we have \(T(r,g)\leq (2+\varepsilon) T(r,f)+S(r,f)\) where \(S(r,f)= o\{T(r,f)\}\), as \(r\to \infty\) possibly outside a set of finite linear measure.