A cosine inequality in the hyperbolic geometry
type: Item
instance of: scholarly article
title: A cosine inequality in the hyperbolic geometry
zbMATH Open document ID: 1203.53014
author: J. Martínez
published in: Applied Mathematics Letters
publication date: 2010-06-25T00:00:00Z
review text: The authors consider a hyperbolic domain \(D\) in the \(n\)-dimensional Euclidean space \(\mathbb R^n\). They define the hyperbolic length and then the hyperbolic distance \(h_D(z_1, z_2)\) from \(z_1\) to \(z_2\). The minimal hyperbolic length path is called the hyperbolic geodesic \(\gamma_h[x,y]\). Following \textit{F. W. Gehring} and \textit{B. P. Palka} [J. Anal. Math. 30, 172--199 (1976; Zbl 0349.30019)], they introduce the quasihyperbolic metric and then the quasihyperbolic geodesic \(\gamma_k[x,y]\). Furthermore, they consider the hyperbolic trigon \(T\) with vertices \(x, y , z\), formed by \(\gamma_h[x,y], \gamma_h[y,z], \gamma_h[z,x]\) resp. \(\gamma_k[x,y], \gamma_k[y,z], \gamma_k[z,x]\). The interior of the hyperbolic geodesic trigon is the set of points in \(D\) that is enclosed by the hyperbolic geodesic trigon. The quasihyperbolic geodesic trigon is constructed analogously. If the interior of such a trigon \(T\) is simply connected then \(T\) is called a triangle \(\Delta_h(x,y,z)\) resp. \(\Delta_k(x,y,z)\). Otherwise \(T\) is called a trigon \(\Delta_h^*(x,y,z)\) resp. \(\Delta_k^*(x,y,z)\). \textit{R. Klén} [Helsinki (2009), \texttt{http://acadsci.fi/mathematica/e-theses/index.html}] considered the law of cosines in hyperbolic geometry and quasihyperbolic geometry. He obtained the following results: If \(D = \mathbb R^2 \setminus \{0\}\) and \(x,y,z \in D\) then for the quasihyperbolic triangle \(\Delta_k(x,y,z)\) holds \[ k^2_D (x,y) = k^2_D (x,z) + k^2_D (y,z) -2k_D(x,z) k_D (y,z) \cos \angle_k (y,z,x). \] For \(\Delta^*_k(x,y,z)\) holds \[ k^2_D (x,y) = k^2_D (x,z) + k^2_D (y,z) -2k_D(x,z) k_D (y,z) \cos \angle_k (y,z,x) - 4 \pi (\pi -\alpha) \] with \(\alpha = \angle(x,0,y)\). A conjecture of Klén concerning a cosine inequality for any hyperbolic domain \(\mathbb H\) can be proved by the authors: They show that for three distinct points \(x, y, z\) of \(\mathbb H^2\) holds \(k^2_{\mathbb H^2}(x,y) \geq k^2_{\mathbb H^2} (x,z) + k^2_{\mathbb H^…
Mathematics Subject Classification ID: 53B20
Mathematics Subject Classification ID: 51M10
Mathematics Subject Classification ID: 51M09
zbMATH DE Number: 5726676
zbMATH Keywords: hyperbolic metric
zbMATH Keywords: quasihyperbolic metric
zbMATH Keywords: hyperbolic geodesic
zbMATH Keywords: quasihyperboloic geodesic
zbMATH Keywords: trigon
zbMATH Keywords: triangle
zbMATH Keywords: law of cosines
zbMATH Keywords: cosine inequality
reviewed by: Johannes Böhm
MaRDI profile type: Publication
full work available at URL: j.aml.2010.04.004
OpenAlex ID: W2009813738
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DOI: 10.1016/J.AML.2010.04.004
description: scientific article; zbMATH DE number 5726676