Geometric properties of a new hyperbolic type metric

A new distance function $\tilde{S}_{G,c}$ in metric space $(X,d)$ is introduced as \begin{align*} &\tilde{S}_{G,c}(x,y)=\log{\left(1+\frac{cd(x,y)}{\sqrt{1+d(x)}\sqrt{1+d(y)}}\right)} \end{align*} for $x$, $y\in X$ and $c$ is an arbitrary positive real number. We find that $\tilde{S}_{G,c}$ is a metric for $c\ge 2$. In general, the condition $c\geq2$ can not be improved. In this paper we investigate some geometric properties of the metric $\tilde{S}_{G,c}$ including the comparison inequalities between this metric and the triangular ratio metric and the inclusion relation between some metric balls. We show the quasiconformality of a bilipschitz mapping in metric $\tilde{S}_{G,c}$ and the distortion property of the metric $\tilde{S}_{\partial\mathbb{B}^n,c}$ under M\"obius transformations of the unit ball.