On a conjecture of W. Bergweiler
\textit{W.Bergweiler} [Arch. Math., 64, 199--202 (1995; Zbl 0818.30021)] conjectured that if \(f\) is a transcendental meromorphic function which is not of the form \(f= \exp (\alpha z + \beta)\) and if \(a\neq 1\) and \(a\neq (n+1)/n\), then \(ff'' - a(f')^{2}\) has at least one zero, where \(n\) is any integer. In response to this conjecture the authors of the paper prove that if \(ff'' - a(f')^{2}\) has only finitely many zeros then \(f\) is of the form \(f = R\exp(P)\) for a rational function \(R\) and a polynomial \(P\), where \(a\neq 1, (n\pm 1)/n\) for any integer \(n\).
scientific article; zbMATH DE number 2024720