  An improved version of quasiinvariance property of the quasihyperbolic metric
under M\"obius transformations of the unit ball in ${\mathbb R}^n, n \ge 2,$ is
given. Next, a quasiinvariance property, sharp in a local sense, of the
quasihyperbolic metric under quasiconformal mappings is proved. Finally,
several inequalities between the quasihyperbolic metric and other commonly used
metrics such as the hyperbolic metric of the unit ball and the chordal metric
are established.
