The spherical-distance formula for the rational-map energy functional

Let $\operatorname{Rat}_k$ be the space of rational self-maps of $\mathbb{CP}^1$ of degree $k$, represented by $f(z)=p(z)/q(z)$ with roots $p_i$ of $p$ and roots $q_j$ of $q$. Let $\rho(p_i,q_j)$ denote the spherical distance between the corresponding points of $\mathbb{CP}^1$. The functional $\mathcal{A}(f)$ is defined by

$$\mathcal{A}(f)=\log\frac{\sup_{z\in\mathbb{C}}\left(|z|^2+1\right)^{-k}\left(|p(z)|^2+|q(z)|^2\right)}{\inf_{z\in\mathbb{C}}\left(|z|^2+1\right)^{-k}\left(|p(z)|^2+|q(z)|^2\right)}.$$

Spherical-distance formula. The functional $\mathcal{A}(f)$ agrees, up to scale or something...?, with

$$\log\frac{\max_{i\neq j}\rho(p_i-q_j)}{\min_{i\neq j}\rho(p_i-q_j)}.$$

Here the indices range over the roots appearing in the representation of $f$. This is presented as a proposed identification of the energy functional with a root-separation expression; the statement is informal and its precise normalization and even the notation $\rho(p_i-q_j)$ require clarification.