One-point hyperbolic-type metrics

We study basic properties of one-parametric families of the j -metric, the scale-invariant Cassinian metric and the half-Apollonian metric on locally compact, noncomplete metric spaces. We first establish basic properties of these metrics on once-punctured general metric spaces and obtain sharp estimates between these metrics, and then we show that all these properties, except for δ -hyperbolicity, extend to the settings of locally compact noncomplete metric spaces. We also show that these metrics are δ -hyperbolic only if the underlying space is a once-punctured metric space.