Normal families and fixed points of iterates

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{F} $$\end{document} be a family of holomorphic functions and suppose that there exists ɛ > 0 such that if f ∈ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{F} $$\end{document}, then |(f2)′(ξ)| ⩽ 4 − ɛ for all fixed points ξ of the second iterate f2. We show that then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{F} $$\end{document} is normal. This is deduced from a result which says that if p is a polynomial of degree at least 2, then p2 has a fixed point ξ such that |(p2)′(ξ)| ⩾ 4. The results are motivated by a problem posed by Yang Lo.