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An improved version of the quasiinvariance property of the quasihyperbolic metric under Möbius transformations of the unit ball in Rn, n≥ 2, is given, and 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.