Lipschitz conditions, triangular ratio metric, and quasiconformal maps

Abstract

Abstract. The triangular ratio metric is studied in subdomains of the complex plane and Euclidean n-space. Various inequalities are proven for this metric. The main results deal with the behavior of this metric under quasiconformal maps. We also study the smoothness of metric disks with small radii.

Geometric properties of the Cassinian metric

“…, [12]. The choice M(α, β) = α + β similarly leads to the triangular ratio metric recently investigated in [4], [10]. We refer to the next section for the explicit definition of the Cassinian metric.…”

“…If , then the corresponding relative metric defines the Cassinian metric introduced in and subsequently studied in , . The choice similarly leads to the triangular ratio metric recently investigated in , . We refer to the next section for the explicit definition of the Cassinian metric.…”

Barrlund's distance function and quasiconformal maps

“…By [5,Corollary 3.30] and Theorem 3.9 (extended to include the case p = ∞), we obtain We also recall some notation about special functions and the fundamental distortion result of quasiregular maps, a variant of the Schwarz lemma for these maps. For r ∈ (0, 1) and K > 0, we define the distortion function…”

“…In the case p = 1 Conjecture 4.4 was formulated in [5] and it was shown in [5, Thm 1.5] that R(1, a) ≥ 1 + |a| . We now extend this last inequality for all p .…”

Some Remarks on the Visual Angle Metric

“…The triangular ratio metric and the hyperbolic metric satisfy the following inequality in the unit ball [CHKV,Lemma 3.4,Lemma 3.8] and [CHKV,Theorem 3.22]:…”