  We study a new generalized version of the point pair function defined with a
constant $\alpha>0$. We prove that this function is a quasi-metric for all
values of $\alpha>0$, and compare it to several hyperbolic-type metrics, such
as the $j^*$-metric, the triangular ratio metric, and the hyperbolic metric.
Most of the inequalities presented here have the best possible constants in
terms of $\alpha$. Furthermore, we research the distortion of the generalized
point pair function under conformal and quasiregular mappings.
