Normal families and shared values of meromorphic functions. II. (Q1848433)

Let \(F\) be a family of functions meromorphic in a domain \(D\), let \(k\) be a positive integer and let \(b,c,d\) be nonzero complex numbers such that \(b\not=d\). Suppose that for all \(f\in F\) the zeros of \(f\) have multiplicity at least \(k\). Suppose also that \(f^{(k)}(z)=b\) whenever \(f(z)=0\) and that \(f(z)=c\) whenever \(f^{(k)}(z)=d\). It is shown that \(F\) is normal if either \(k\geq 3\), or \(k=2\) and \(b\not= 4d\), or \(k=1\) and \(b\not=(m+1)d\) for all positive integers \(m\). This result extends various other results obtained previously by the authors and others. Interesting examples show that the extra conditions in the case \(k=2\) and \(k=1\) are needed. The main tool used is a rescaling lemma introduced by L. Zalcman, later extended by X. C. Pang, and then further refined by \textit{X. C. Pang} and \textit{L. Zalcman} [Bull. Lond. Math. Soc. 32, 325--331 (2000; Zbl 1030.30031)]. A second result concerns the normality of families of functions \(f\) for which \(f(z)f^{(k)}(z)=a\) whenever \(f^{(k)}(z)=b\), and vice versa.