Carathéodory metric

Remarks

$c_X$ is a pseudodistance, as a supremum of pseudodistances $(p, q) \mapsto \rho(f(p), f(q))$. If $g \colon X \to Y$ is a holomorphic map, then $c_Y(g(p), g(q)) \le c_X(p, q)$, since $f \circ g$ is admissible for $X$ whenever $f$ is admissible for $Y$. Example: $c_{\mathbb{D}} = \rho$; the identity gives $c_{\mathbb{D}} \ge \rho$, and the Schwarz–Pick theorem gives $\rho(f(a), f(b)) \le \rho(a, b)$ for every $f$. Example: if $X$ is a bounded domain in $\C^n$, then $c_X$ is a distance, since a suitable linear function, rescaled, maps $X$ into $\mathbb{D}$ and separates any two given points. Non-example: $c_{\C^n} \equiv 0$, since every bounded holomorphic function on $\C^n$ is constant. For every connected $X$, $c_X \le k_X$, where $k_X$ is the Kobayashi pseudodistance.