Differential polynomials that share three finite values with their generating meromorphic function
Making use of standard notations of shared value problems for meromorphic functions, \textit{G. Gundersen} [J. Math. Anal. Appl. 75, 441-446 (1980; Zbl 0447.30018)] and \textit{E.~Mues} and \textit{N.~Steinmetz} [Manuscr. Math. 29, 195-206 (1979; Zbl 0416.30028)] proved that \(f=f'\) provided that \(f\) is non-constant meromorphic and \(f\) and~\(f'\) share three finite distinct values \(b_1\), \(b_2\), \(b_3\) IM. \textit{G.~Frank} and \textit{W.~Schwick} succeeded to replace \(f'\) with \(f^{(k)}\), \(k\in{\mathbf N}\) [see Results Math. 22, No. 3/4, 679-684 (1992; Zbl 0763.30011)], while \textit{E.~Mues} and \textit{M.~Reinders} extended the result to \[ L(f):=a_kf^{(k)}+a_{k-1}f^{(k-1)}+\dots+a_0f,\qquad a_0,\dots a_k\in\mathbb{C},\quad a_k\neq 0, \] under the restriction \(2\leq k\leq 50\) [Result. Math. 22, 725-738 (1992; Zbl 0763.30012)]. The present paper removes the restriction by proving that \(f=L(f)\), whenever \(k\geq 2\) and \(f\) and \(L(f)\) share three finite distinct values \(b_1\), \(b_2\), \(b_3\) IM. The proof applies known ideas from shared value theory, combined with some ideas in [Result. Math. 22, No. 3/4, 725-738 (1992; Zbl 0763.30012] plus a careful analysis of the value distribution of~\(f\).
scientific article; zbMATH DE number 1548294