Taylor Coefficients of Negative Powers of Schlicht Functions

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\cal S$\end{document} denote the class of normalized schlicht functions in the unit disk. We consider for f ∊ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\cal S}$\end{document} and λ < 0 the Taylor coefficients an (λ, f) of (f(z)/z)λ and prove that ∣an(λ, f)∣ ≤ ∣an(λ, k)∣ for every f ∈ S and every 1 ≤ n ≤ − λ + 1, where k(z) = z(l − z)−2 is the Koebe function. We also give a necessary condition such that the Koebe function maximizes the functional \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum_{k=1}^n \sigma_k \mid a_k(\lambda,f) \mid^2$$\end{document} in the class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\cal S}$\end{document} for given weights σk ∈ R. These results supplement and complement previous results due to de Branges, Hayman and Hummel and others. Our proofs are based on the Löwner differential equation combined with optimal control methods.