Normal families concerning shared values
Let \(F\) be a family of analytic functions in the unit disk \(D\). Suppose there is a nonzero complex number \(a\) such that for each \(f\) in \(F\), \(f\), \(f'\), and \(f''\) all share the value \(a\) ignoring multiplicity. Using Zalcman's principle the authors show \(F\) is a normal family. Since the family \(G= \{f_n|f_n(z)= \exp(nz)\) for \(n=1,2,\dots\}\) is not normal, the value \(a\) cannot be zero. Related results appear in \textit{W. Schwick} [Arch. Math. 59, No. 1, 50-54 (1992; Zbl 0758.30028)] and \textit{G. Jank}, \textit{E. Mues} and \textit{L. Volkmann} [Complex Variables, Theory Appl. 6, 51-71 (1986; Zbl 0603.30037)].
scientific article; zbMATH DE number 1436114