Uniqueness of meromorphic functions and a question of K. Tohge
The paper under review concerns with the uniqueness of meromorphic functions sharing three values. \textit{K. Tohge} showed that if two meromorphic functions, with hyper-order less than one, share \(0, 1, \infty\) {CM} and their first order derivatives share a non-zero finite complex number {CM}, then the two functions satisfy one of three very special equations. He asked in the paper that if the condition {CM} can be weakened [``Meromorphic functions covering certain finite sets at the same points'', Kodai Math. J. 11, 249--279 (1988; Zbl 0663.30024)]. The current authors consider the uniqueness problem on three weighted shared values. Denote by \(E_k(a;f)\) the set of all \(a\)-points of \(f\) where an \(a\)-point of multiplicity \(m\) is counted \(m\) times if \(m \leq k\) and \(k+1\) times if \(m > k\). Write \(f, g\) share \((a, k)\) to mean that \(f,g\) share a value \(a\) with weight \(k\), that is \(E_k(a;f) =E_k(a;g)\). We note that \(f,g\) share a value \(a\) {IM} or {CM} if and only if \(f,g\) share \((a,0)\) or \((a, \infty)\). The authors prove that if two distinct transcendental meromorphic functions sharing \((0,1), (1,m),(\infty, k)\), where \((m-1)(mk-1)>(1+m)^2\) and moreover, if \(f', g'\) share \(a\) {CM}, where \(a\) is a non-zero finite complex number, then \(f,g\) share \(0,1\), and \(\infty\) {CM}.
scientific article; zbMATH DE number 5073026