Comparison theorems for hyperbolic type metrics

Abstract

Abstract. The connection between several hyperbolic type metrics is studied in subdomains of the Euclidean space. In particular, a new metric is introduced and compared to the distance ratio metric.

A new intrinsic metric and quasiregular maps

“…We will now prove that the expression on the left hand side of the inequality of Lemma 3.3 satisfies the triangle inequality and for that purpose we need the following refined form of Proposition 3.2 for c ≥ 1 . This refined result and some of the lower bounds that will be proved below for the function F c in Proposition 3.2, also lead to improved constants in some of the results of [6].…”

“…Here we study a function recently used as a tool by O. Dovgoshey, P. Hariri, and M. Vuorinen [6] and show that this function satisfies the triangle inequality and, indeed, defines an intrinsic metric of a domain. Moreover, we compare it to the distance ratio metric and find two-sided bounds for it.…”

“…Distance functions specific to a domain G ⊂ R n , n ≥ 2 , or, as, we call them, intrinsic metrics, are some of the key notions of geometric function theory and are currently studied by many authors. See for instance the recent monographs [8,9,10,15] and papers [4,6,7,11,12,13]. In [8] intrinsic metrics are used as a powerful tool to analyse the properties of quasidisks and [10] provides a survey of some recent progress in the field.…”

“…In this paper our goal is to prove that the expression (1.2) studied in [6] for (X, ρ) = (D, j D ) is, in fact, a metric. We also prove several upper and lower bounds for this new metric.…”

Remarks on the scale invariant Cassinian metric

“…If we can describe the balls of a metric space "explicitly", then we already know a lot about the geometry of the metric -this requires that we can estimate the metric in terms of well-known metrics. For a survey and comparison inequalities between some of these metrics, see [4,7,10,15,16,17,19,20].…”