Schwarz–Pick theorem

Remarks

Proof idea: for $a \in \mathbb{D}$ let $\varphi_a(z) = \frac{z - a}{1 - \bar a z}$, a Möbius transformation of $\mathbb{D}$ onto itself with $\varphi_a(a) = 0$. The map $g = \varphi_{f(w)} \circ f \circ \varphi_w^{-1}$ sends $\mathbb{D}$ into $\mathbb{D}$ and fixes $0$. The Schwarz lemma gives $\abs{g(\zeta)} \le \abs{\zeta}$, which is the first inequality with $\zeta = \varphi_w(z)$; letting $z \to w$ gives the second. In terms of the hyperbolic metric $\frac{2 \abs{dz}}{1 - \abs{z}^2}$ of the Poincaré disk model, the theorem says that $f$ does not increase hyperbolic distances. The Möbius transformations of $\mathbb{D}$ onto itself are exactly the hyperbolic isometries that preserve orientation. Example: $f(z) = z^2$ gives $\abs{2z}/(1 - \abs{z}^4) \le 1/(1 - \abs{z}^2)$, that is $2\abs{z} \le 1 + \abs{z}^2$, which is $(1 - \abs{z})^2 \ge 0$.

Going deeper

For a student

The theorem shows that the hyperbolic metric is the natural metric for holomorphic maps of the disc. Every holomorphic self-map is a contraction for it, and the automorphisms are the isometries. This point of view leads to the Ahlfors–Schwarz lemma for metrics of negative curvature and to the Kobayashi metric in several complex variables. It also gives quick proofs of distortion estimates, for example $\abs{f'(0)} \le 1 - \abs{f(0)}^2$.