On the value distribution of \(f +a(f\prime)^n\)
Let \(f\) be a transcendental meromorphic function, and let \(a\) be a nonzero finite complex number. \textit{Y. Ye} [Chin. Ann. Math., Ser. B 15, No. 1, 75--80 (1994; Zbl 0801.30027)] proved that \(f+a(f')^n\) assumes every finite complex value infinitely often for each positive integer \(n\geq 3\). It is a question whether the result remains valid for \(n=2\). The authors give an affirmative answer for this question. The authors also consider a normality criterion corresponding to the result above. Let \({\mathcal F}\) be a family of meromorphic functions on the plane \(D\), let \(n\geq 2\) be a positive integer, and let \(a\neq 0\), \(b\) be complex numbers. If, for each \(f\in{\mathcal F}\), all zeros of \(f\) are multiple and \(f+a(f')^n\neq b\) on \(D\), then \({\mathcal F}\) is normal on \(D\). The tools for the proofs are the standard Nevanlinna theory and the rescaliug lemma due to the second author of this paper.
scientific article; zbMATH DE number 5346104