We study basic properties of one-parametric families of the
-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
-hyperbolicity,
extend to the settings of locally compact noncomplete metric spaces. We also show that these metrics
are
-hyperbolic
only if the underlying space is a once-punctured metric space.
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