Mathematische Zeitschrift

, Volume 259, Issue 1, pp 187–196 | Cite as

Estimation of the hyperbolic metric by using the punctured plane

Article

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 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.

Keywords

Hyperbolic metric Punctured plane Elliptic integral 

Mathematics Subject Classification (2000)

Primary 30F45 

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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of MathematicsAristotle University of ThessalonikiThessalonikiGreece

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