In [1], Beardon introduced the Apollonian metric α defined for any domain D in ℝ by

formula here

This metric is Möbius invariant, and for simply connected plane domains it satisfies the inequality αD[les ]2ρD, where ρD denotes the hyperbolic distance in D, and so gives a lower bound on the hyperbolic distance. Furthermore, it is shown in [1, Theorem 6.1] that for convex plane domains, the Apollonian metric satisfies ρD[les ]4 sinh [½αD], and, by considering the example of the infinite strip {x+iy[ratio ][mid ]y[mid ]<1}, that the best possible constant in this inequality is at least π. In this paper we make the following improvements.