Meromorphic functions that share four or five pairs of values (Q1653341)

Let \(a,b\in\mathbb{C}\cup\{\infty\}\) and let \(f\) and \(g\) be meromorphic functions in the complex plane. It is said that \(f\) and \(g\) share the pair of values \((a,b)\) if \(f(z)=a\) if and only if \(g(z)=b\). In addition, it is said that \(f\) and \(g\) share the pair of values \((a,b)\) counting multiplicities, if, for all \(z\), the multiplicity of \(f(z)=a\) is the same as the multiplicity of \(g(z)=b\). In the paper under review the authors show that if two non-constant meromorphic functions \(f\) and \(g\) share five pairs of values, out of which at least two pairs are shared counting multiplicities, then \(f\) is a Möbius transformation of \(g\). Examples are provided showing that this result is sharp.