Unicity theorems for meromorphic functions that share three values (Q1906560)

Two meromorphic functions \(f\) and \(g\) share the complex value \(a\) if \(f(z)=a\) implies \(g(z)=a\) and vice versa. The value \(a\) is shared CM (counting multiplicities) if, in addition, \(f\) and \(g\) have the same multiplicities at each \(a\)-point. Let \(f\) and \(g\) be distinct non-constant meromorphic functions in the complex plane sharing the value \(0\), \(1\) and \(\infty\) CM. The main result of the paper under review is Theorem 4: If \[ N(r,1/(f-a))\neq T(r,f)+ S(r,f)\tag{\(*\)} \] then \(f\) and \(g\) are related by one of the following relations: \((f-a)(g+a-1)= a(1-a)\), \(f+(a-1)g=a\) or \(f=ag\). \textit{G. Brosch} [Eindeutigkeitssätze für meromorphe Funktionen, Dissertation, RWTH Aachen (1989; Zbl 0694.30027)] already proved (Folgerung 5.2, which is a corollary to a result of E. Mues) that the condition \((*)\) implies \(f=L(g)\) with a Möbius transformation \(L\). The conclusion of Theorem 4 then follows immediately. The proof in the present paper is almost identical to the proof of G. Brosch.