Inequalities of generalized hyperbolic metrics (Q2477954)

For a quadruple \(a,b,c,d\) of distinct points in \( \overline{\mathbb{R}}^n = \mathbb{R}^n \cup \{ \infty\} \) the absolute crossratio is \[ | a,b,c,d| = q(a,c)q(b,d)/(q(a,b)q(c,d))\,, \] where \(q\) denotes the chordal metric. The basic property of the absolute crossratio is invariance under Möbius transformations. For a subdomain \(G \subset \overline{\mathbb{R}}^n \) \textit{P. Seittenranta} [Math. Proc. Camb. Philos. Soc. 125, No. 3, 511--533 (1999; Zbl 0917.30015)] introduced the Möbius invariant metric \[ \delta_G(x,y) = \sup_{a,b\in \partial G} \log(1+ | a,x,b,y| ) \] and studied its quasiinvariance under quasiconformal mappings. Continuing his earlier work [J. Math. Anal. Appl. 301, No. 2, 336--353 (2005; Zbl 1069.54019), J. Math. Anal. Appl. 274, No. 1, 38--58 (2002; Zbl 1019.54011)] the present author considers, in addition to \(\delta_G\), several other Möbius invariant metrics. One of these is \[ \rho_G(x,y)= \sup_{a,b\in \partial G} {\mathrm{cosh}}^{-1}(1+ | a,x,b,y| | a,y,b,x| /2)\,. \] The author proves many inequalities these for these metrics. For instance, the following result is proved with \(c= {\mathrm{cosh}}^{-1}(3) /{\mathrm{log}}(3)\) \[ \delta_G \leq \rho_G \leq c \delta_G \,. \]