Geometric properties of the cassinian metric

Abstract

In this thesis we obtain various inequalities between the so-called Cassinian metric and other well-known hyperbolic-type metrics. For this, comparison of a scale invariant Cassinian metric with a Gromov hyperbolic metric and other hyperbolic-type metrics plays a significant role. We discuss the local convexity property of the Cassinian metric balls and their inclusion relations with other hyperbolic-type metric balls by fixing centre common to each pair of metric balls. The metric ball inclusion properties of the scale invariant Cassinian metric and the Gromov hyperbolic metric with other hyperbolic-type metric balls are also interpreted. We study the distortion property of the Cassinian metric under M ̈obius transformations of the unit ball onto itself and under M ̈obius transforma- tions of a punctured ball onto another punctured ball. Hence, the distortion property of the scale invariant Cassinian metric under M ̈obius transformations of a punctured ball onto another punctured ball is also natural to discuss. We also discuss the quasi-invariance property of the scale invariant Cassinian metric and the Gromov hyperbolic metric under quasiconformal mappings of R n , n ≥ 2. Finally, we estimate the modulus of continuity for the identity mapping on a bounded domain equipped with the Cassinian metric onto the same domain with the Euclidean metric.