  We study a hyperbolic type metric $h_{G,c}$ introduced by Dovgoshey, Hariri,
and Vuorinen. We find the best constant $c>0$, for which this function
$h_{G,c}$ is a metric in specific choices of $G$. We give several sharp
inequalities between $h_{G,c}$ and other hyperbolic type metrics, and also
offer a few results related to ball inclusion.
