Inequalities for the generalized point pair function

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.