= Schwarz-Pick theorem
{c}
{title2=$\rho(f(z),f(w))\leq\rho(z,w)$}
= Schwarz-Pick lemma
{c}
{synonym}
A holomorphic self-map of the <unit disc> contracts its pseudohyperbolic distance. Conjugating by <Möbius transformations> that send one chosen source point and its image to zero reduces the statement to the <Schwarz lemma>. In the <complex upper half-plane> this gives
$$
\left|\frac{f(z)-f(w)}{f(z)-\overline{f(w)}}\right|\leq\left|\frac{z-w}{z-\overline w}\right|.
$$
If $f(iY)=iY+o(Y)$ at infinity, multiply the equivalent inequality between one minus the squared distances by $Y$ and let $Y\to\infty$. It yields $\operatorname{Im}f(z)\geq\operatorname{Im}z$. Applied to the inverse <mapping-out function> of a <compact H-hull>, this proves the nonnegativity of its <half-plane capacity>.
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