Bounds for derivatives of hyperbolic type distances

Let $G$ be a starlike domain containing the origin, let $d_G(0)$ denote the Euclidean distance from the origin to the boundary of $G$, and let $m$ be a hyperbolic type distance with associated function $f_m$. Bounds conjecture. There exist constants $a\geq 0$ and $b\geq 1$ such that

$$\frac{a}{d_G(0)}\leq f_m'(0)\leq\frac{b}{d_G(0)}.$$

The preceding examples show that the derivative has this form for standard hyperbolic distances in shifted half-spaces and balls, while the paper notes that the sharper universal equality suggested earlier is false in general. The asserted two-sided bound is presented as the statement supported by the authors' study.