In this paper we prove a sharp distortion property of the Cassinian
metric under Möbius transformations of a punctured ball onto another
punctured ball. The paper also deals with discussion on local convexity
properties of the Cassinian metric balls in proper subdomains of .
Inclusion properties of the Cassinian metric balls with other
hyperbolic-type metric balls are also investigated. In particular,
several conjectures are also stated in response to sharpness of the
inclusion relations.