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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.

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Authors and Affiliations

  1. Department of Mathematics, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece

    Dimitrios Betsakos

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  1. Dimitrios Betsakos
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Correspondence to Dimitrios Betsakos.

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

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Betsakos, D. Estimation of the hyperbolic metric by using the punctured plane. Math. Z. 259, 187–196 (2008). https://doi.org/10.1007/s00209-007-0218-0

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  • DOI: https://doi.org/10.1007/s00209-007-0218-0

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