Inequalities for the generalized point pair function

We study a new generalized version of the point pair function defined with a constant α > 0. We prove that this function is a quasi-metric for all values of α > 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 α. Furthermore, we research the distortion of the generalized point pair function under conformal and quasiregular mappings.