On the zeros of a homogeneous differential polynomial
It was proved by \textit{E. Mues} [Manuscripta Math. 23, 325-341 (1978; Zbl 0382.30021)] that \(g(z)= e^{\alpha z+\beta}\) is the only transcendental entire function such that \(W_ a:= gg''- ag'{}^ 2\) is zero-free in the plane for some constant \(a\neq 1\). The author of the present paper considers meromorphic functions \(g\) and obtains several results of the following type: Theorem 1. Let \(g\) be a non-constant meromorphic function and assume that \(W_ a\) does not vanish identically for some \(a\neq 1\). Then \(T(r,{g'\over g})\leq \alpha_ a\overline N(r,{1\over W_ a})+ \beta_ a\overline N(r,{1\over g})+ V_ a(r)\), where \(\alpha_ a\), \(\beta_ a\) are explicit constants depending on \(a\), and \(V_ a\) is an error term. A similar result holds for \(a= 1\), except when \(g(z)= C_ 1 e^{\lambda_ 1 z}+ C_ 2 e^{\lambda_ 2 z}\) \((\lambda_ 1\neq \lambda_ 2\), \(C_ 1C_ 2\neq 0)\).
scientific article; zbMATH DE number 558886