The precise bound for the area–length ratio in Ahlfors’ theory of covering surfaces

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$S=\overline{\mathbb{C}}$\end{document} be the unit Riemann sphere. A basic consequence of Ahlfors’ theory of covering surfaces is that there exists a positive constant h such that for any simply-connected surface Σ over S∗=S∖{0,1,∞}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A(\varSigma)\leq hL(\partial\varSigma).$$\end{document} Here A(Σ) is the area of Σ and L(∂Σ) is the length of the boundary of Σ. The goal of this paper is to prove that the least possible value of h is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{0}=\max_{\theta\in\lbrack0,\pi/2]} \biggl[ \frac{ ( \pi+\theta ) \sqrt{1+\sin^{2}\theta}}{\arctan\frac{\sqrt{1+\sin^{2}\theta}}{\cos\theta}}-\sin\theta \biggr] =4.03415979051\ldots $$\end{document}