Normal families of meromorphic functions whose derivatives omit a function
The authors prove three theorems including the following interesting generalization of earlier results of \textit{L. Yang} [Sci. Sin., Ser. A 29, 1263--1274 (1986; Zbl 0629.30032)] and \textit{X. Pang} and \textit{L. Zalcman} [Isr. J. Math. 136, 1--9 (2003; Zbl 1040.30016)]: Let \(F\) be a family of functions meromorphic on a domain \(D\), all of whose zeros have multiplicity at least \(m+3\). Let \(h\) be an analytic function in \(D\) which is not identically zero, and for each \(f\) in \(F\), \(f^{(m)}(z)\neq h(z)\) for \(z\in D\). Then \(F\) is a normal family on \(D\). An example shows \(m+3\) cannot be replaced by \(m+2\) without some additional change in hypothesis for example if all zeros of \(h\) are multiple. Proofs rely on a local version of lemma 2 in [\textit{X. Pang} and \textit{L. Zalcman}, Bull. Lond. Math. Soc. 32, 325--331 (2000; Zbl 1030.30031)] describing when a family is not normal, and a reduction to consideration of the behavior of rational functions arrived at thorough use of results of \textit{W. Bergweiler} [Arch. Math. 64, 199--202 (1995; Zbl 0818.30021)] and \textit{W. Bergweiler} and \textit{A. Eremenko} [Rev. Mat. Iberoam. 11, 355--373 (1995; Zbl 0830.30016)].
scientific article; zbMATH DE number 1995505