, Volume 259, Issue 1, pp 187-196

Estimation of the hyperbolic metric by using the punctured plane

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Abstract

Let $\rho_\Omega$ denote the density of the hyperbolic metric for a domain Ω in the extended complex plane $\overline{\mathbb {C}}$ . We prove the inequality $$\rho_{\Omega}(z)\leq C\, {\rm sup} \{\rho_{\mathbb {C}\setminus \{a,b\}}(z): a,b\in\partial \Omega\},\quad z\in \Omega,\,\Omega\subset \mathbb {C},$$ 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 $$\rho_\Omega(z)\leq C_1\, {\rm sup} \{\rho_{\overline{\mathbb {C}}\setminus \{a,b,c\}}(z): a,b,c\in\partial \Omega\},\quad z\in \Omega$$ with an unspecified constant C 1. 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.

The author was partially supported by the EPEAK programm Pythagoras II (Greece).