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In this paper, we consider a scale invariant Cassinian metric and a Gromovhyperbolic metric. We discuss a distortion property of the scale invariantCassinian metric under M\"obius maps of a punctured ball onto another puncturedball. We obtain a modulus of continuity of the identity map from a domainequipped with the scale invariant Cassinian metric (or the Gromov hyperbolicmetric) onto the same domain equipped with the Euclidean metric. Thequasi-invariance properties of both the metrics under quasiconformal maps arealso established.

Mapping properties of a scale invariant Cassinian metric and a Gromov hyperbolic metric