A normality criterion for meromorphic functions (Q418263)

Let \(a\) be a nonzero complex number and \({\mathcal F}\) a family of meromorphic functions defined in a domain \(\mathbb{D}\). In the case that \(k\) and \(n\) are positive integers such that \(n\geq 1\) if \(k= 1\) and \(n\geq 2\) if \(k\geq 2\), and assuming that \(f^n(f^{k+1})^{(k)}\neq a\) for every \(f\) in \({\mathcal F}\), the authors prove that \({\mathcal F}\) is a normal family. The result complements a theorem by \textit{Y. Li} and \textit{Y. Gu} in [J. Math. Anal. Appl. 354, No. 2, 421--425 (2009; Zbl 1169.30013)] where monomials of the form \((f^n)^{(k)}\) for \(n\geq k+ 2\) are considered.