A cosine inequality in the hyperbolic geometry
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 \(k^2_{\mathbb H^2}(x,y) \geq k^2_{\mathbb H^2} (x,z) + k^2_{\mathbb H^2} (y,z) - 2k_{\mathbb H^2}(x,z) k_{\mathbb H^2} (y,z) \cos \angle_k (y,z,x)\). Moreover, by means of an example, they show that strict inequality can occur.
scientific article; zbMATH DE number 5726676