Triangular ratio metric in the unit disk

Abstract

The triangular ratio metric is studied in a domain $G⊊{\mathbb{R}}^n$, $n\ge 2$. Several sharp bounds are proven for this metric, especially in the case where the domain is the unit disk of the complex plane. The results are applied to study the Hölder continuity of quasiconformal mappings.

Keywords:

• Hölder continuity
• hyperbolic geometry
• midpoint rotation
• quasiconformal mappings
• triangular ratio metric

AMS Subject Classifications:

• Primary: 51M10
• Secondary: 30C65

Acknowledgements

The authors are indebted to Professor Masayo Fujimura and Professor Marcelina Mocanu for their kind help in connection with the proof of Theorem 4.4.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

The research of the first author was supported by Finnish Concordia Fund.