Periodic points and normal families concerning multiplicity
Let \(D\) be a domain in \(\mathbb{C}\) and \(l\) and \(k\) be positive integers. Two main theorems are proved: \begin{itemize} \item[(i)] Let \(k\ge 2\) and \(\mathcal{F}\) be a family of holomorphic functions in \(D\) for which each \(f\in\mathcal{F}\) has all zeros of \(f(z)-z\) multiple and \(f^k\) has at most \(k\) distinct fixed points in \(D\), then \(\mathcal{F}\) is normal in \(D\). \item[(ii)] Let \(k\ge 2\) and \(l\ge 4\) for \(k=2\) and \(l\ge 3\) for \(k\ge 3\). If \(\mathcal{F}\) is a family of meromorphic functions in \(D\) such that for each \(f\in\mathcal{F}\), all zeros of \(f(z)- z\) have multiplicity at least \(l\) and \(f^k\) has at most one fixed point in \(D\), then \(\mathcal{F}\) is normal in \(D\). \end{itemize} Examples are given showing the best possible nature of both theorems. For \(k\ge 2\) and \(\mathcal{F}\) a family of holomorphic functions in \(D\) for which every \(f\in\mathcal{F}\) has the property that both \(f\) and its iteration \(f^k\) have no fixed points in \(D\), \textit{M. Essén} and \textit{S. Wu} [Complex Variables, Theory Appl. 37, No. 1--4, 171--178 (1998; Zbl 1054.30524)] proved that \(\mathcal{F}\) is normal. The corresponding theorem for \(\mathcal{F}\) a family of meromorphic functions was shown to be normal by \textit{J. Chang} and \textit{M. Fang} [J. Anal. Math. 95, 389--395 (2005; Zbl 1080.30033)]. The key ingredients in the proof of (ii) are the now classic Zalcman's lemma and a proposition for \(R\) a non-polynomial rational function for which \(l\ge 4\) for \(k=2\) and \(l\ge 3\) for \(k\ge 3\) showing if all zeros of \(R(z)-z\) have a multiplicity of at least \(l\), then \(R^k(z)\) has at least two distinct fixed points \(a_i\) with \(R^j(a_i)\in\mathbb{C}\) for \(0<j<k\), \(i= 1,2\). Proof of the proposition involves highly detailed computations for cases according to the number of fixed points of \(R\).
normal families of meromorphic functions
scientific article; zbMATH DE number 7033413