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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.