Meromorphic functions that share values by simple and double zeros (Q1901608)

Let \(f\) and \(g\) be nonconstant meromorphic functions in the complex plane. The functions \(f\) and \(g\) are said to share a finite complex value \(c\) by simple zeros if the set of simple zeros of \(f- c\) coincides with the set of simple zeros of \(g- c\). We say \(\infty\) is shared by simple zeros if \(f\) and \(g\) have the same simple poles. The authors show that if \(f\) and \(g\) share seven values for simple zeros, then \(f\equiv g\). Extending their work they show that if \(f\) and \(g\) share six values by simple and double zeros, then \(f\equiv g\). Problems of this nature go back to classical work of R. Nevanlinna. A related work is \textit{E. Mues} [Complex Variables, Theory Appl. 12, No. 4, 169-179 (1989; Zbl 0699.30025)].