Estimation of the hyperbolic metric by using the punctured plane

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho_\Omega$$\end{document} denote the density of the hyperbolic metric for a domain Ω in the extended complex plane \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\mathbb {C}}$$\end{document}. We prove the inequality \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho_{\Omega}(z)\leq C\, {\rm sup} \{\rho_{\mathbb {C}\setminus \{a,b\}}(z): a,b\in\partial \Omega\},\quad z\in \Omega,\,\Omega\subset \mathbb {C},$$\end{document}with C = 8.27. The inequality was proved by Sugawa and Vuorinen with C = 10.33. The proof uses monotonicity properties of the hyperbolic metric for the thrice punctured extended plane. Gardiner and Lakic proved the inequality \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho_\Omega(z)\leq C_1\, {\rm sup} \{\rho_{\overline{\mathbb {C}}\setminus \{a,b,c\}}(z): a,b,c\in\partial \Omega\},\quad z\in \Omega$$\end{document}with an unspecified constant C1. We show that the best constant Σ1 in this inequality is between 3.25 and 8.27. We also prove a related conjecture formulated by Sugawa and Vuorinen.