The Gehring-Hayman identity for the diameter of quasihyperbolic geodesics in convex domain. (Chinese. English summary) Zbl 1299.30119

Qian, Weimao; Zhang, Yichi

Pure Appl. Math. 29, No. 3, 241-245, 274 (2013).

Summary: This paper generalizes the Gehring-Hayman inequality for the diameter of the hyperbolic geodesics in the plane Jordan domain to the quasihyperbolic geodesics in the convex domain of \(n\)-dimensional space. Making use of the Möbius transformation and the quasihyperbolic metric, we prove that the diameter of the quasihyperbolic geodesics with the endpoints \(x\) and \(y\) in the convex domain of \(n\)-dimensional space is equal to the Euclidean distance between \(x\) and \(y\). The obtained result is a generalization and improvement of some known results.

MSC:

30F45	Conformal metrics (hyperbolic, Poincaré, distance functions)
30C20	Conformal mappings of special domains
30C65	Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations

Keywords:

convex domain; quasihyperbolic length; quasihyperbolic distance; quasihyperbolic geodesics; Gehring-Hayman inequality