A New Approach to Support Point Theory for the Class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{S}$$\end{document}

In this paper, we introduce a new technique for measuring the hypothetical non-monotonicity of the argument of the vector tracing the omitted arc of a support point of 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}. It has been previously shown that the number of sign changes of (arg w(t))′ on the omitted arc is finite. Here, we derive upper bounds for that number in terms of both the spherical and Schwarzian derivatives. Thus, our innovative approach identifies an inherently interesting connection between support point theory and these derivatives.