On the value distribution of \(f^ l(f^{(k)})^ n\) (Q1330244)

The main result of the present paper is, in a simplified form, as follows: suppose \(f\) is a transcendental meromorphic function in \(\mathbb{C}\) and \(F = f^ \ell (f^{(k)})^ n - 1\) with \(\ell, k, n\) positive integers, \(\ell \leq 2\). Then (i) if \(\ell = 1\) and \(n > 9e + 1\) (hence \(n \geq 26)\) there exists a positive constant \(C\) and a set \(E \subseteq [0, \infty)\) of infinite measure such that \[ T(r,f^{(k)}) \leq C \overline N \left( r, {1 \over F} \right),\quad r \in E; \] (ii) if \(\ell = 2\) and \(k \geq 2\), \(n \geq 2\), then \[ T(r,f) \leq (2 + \eta) \overline N \left( r, {1 \over F} \right) + S(r,f) \] holds for any \(\eta > 0\). In particular, \(F\) has infinitely many zeros. [Reviewer's remark: The context is as follows: If \(f\) is transcendental, then it is known that \(f^ nf'\) assumes every value \(a \neq 0\) infinitely often, provided \(n \geq 1\) in the entire case and \(n \geq 2\) in the meromorphic case. The question whether \(n \geq 1\) suffices has been open for a long time and has only recently (1994) been settled by Bergweiler- Eremenko and Zalcman (preprints)].