The logarithmic derivative and a homogeneous differential polynomial of a meromorphic function
In the present paper the homogeneous differential polynomial \(W_ a : = gg'' - ag'{}^ 2\) is studied, where \(g\) is nonconstant meromorphic in \(\mathbb{C}\) and \(a\) is any constant. This expression has been considered by \textit{E. Mues} [Manuscr. Math. 23, 325-341 (1978; Zbl 0382.30021)] in the entire case: He was able to show that \(W_ a\) is zero free if and only if \(g(z) = e^{\alpha z + \beta}\), except when \(a = 1\). In the meromorphic case treated here the following inequality is proved: \[ T(r,g'/g) \leq A_ a m(r,g'/g) + B_ a m(r, W_ a'/ W_ a) + C_ a \left( \overline N \left( r, {1 \over W_ a} \right) + \overline N (r,g) \right) + U_ a (r). \tag{*} \] \(A_ a\), \(B_ a\), \(C_ a\) are numerical constants depending on \(a\), \(U_ a (r)\) is an error-term, and \((*)\) holds except in two cases as follows: (i) \(a = 1/2\), \(g(z) = \alpha z^ 2 + \beta z + \gamma\), \(\beta^ 2 - 4 \alpha \gamma \neq 0\); (ii) \(a = 1\), \(g(z) = C_ 1 e^{\lambda_ 1z} + C_ 2 e^{\lambda_ 2 z}\), \(\lambda_ 1 \neq \lambda_ 2\), \(C_ 1C_ 2 \neq 0\). From \((*)\) Mues' result can be deduced.
scientific article; zbMATH DE number 558885