On monomials and Hayman's problem (Q1261819)

The following problem is still open: Let \(f\) be a transcendental meromorphic function in the plane. It is true that, for \(n=1\), \(\psi=f^ nf'\) assumes every value, except possibly zero, infinitely often? In case \(n\geq 2\) this is known to be true, and even in the case \(\psi=f^ n(f')^{n_ 1}\cdots\bigl(f^{(k)}\bigl)^{n_ k}\), \(\max n_ j\geq 1\). In the paper under review it is shown, that \(\psi\) assumes every nonzero value infinitely often, provided \(n\geq 1\), \(n_ 1\geq 1\) and \(f\) ahs only `few' zeros. This additional hypothesis makes the reviewer's proof [Math. Z. 176, 255-264 (1981; Zbl 0466.30026)] work also in the case \(n=1\).