Schwarz lemma
Theorem. If \(f:\mathbb D\to\mathbb D\) is holomorphic and \(f(0)=0\), then \(|f(z)|\le |z|\) and \(|f'(0)|\le1\). If equality occurs at a nonzero point or in the derivative, then \(f(z)=e^{i\theta}z\).
Proof by maximum principle
Proof. Define \(g(z)=f(z)/z\) for \(z\ne0\), and set \(g(0)=f'(0)\). The removable singularity theorem makes \(g\) holomorphic. On \(|z|=r\), \(|g(z)|\le1/r\); letting \(r\uparrow1\) gives \(|g|\le1\).
Automorphisms of the disc
Theorem. Every biholomorphic automorphism of \(\mathbb D\) has the form \[ e^{i\theta}\frac{z-a}{1-\overline a z},\qquad |a|<1. \] Compose with the automorphism sending \(a\) to \(0\), then apply Schwarz's lemma.
Schwarz-Pick estimate
Theorem. Holomorphic maps \(f:\mathbb D\to\mathbb D\) contract the pseudohyperbolic metric: \[ \left|\frac{f(z)-f(w)}{1-\overline{f(w)}f(z)}\right|\le \left|\frac{z-w}{1-\overline w z}\right|. \]
Fixed points
Corollary. A non-identity automorphism of the disc has tightly constrained fixed points. Schwarz's lemma shows that an automorphism fixing \(0\) is a rotation.
Geometric meaning
Remark. Schwarz-Pick turns holomorphic self-maps into contractions for the intrinsic geometry of the disc. This is the beginning of complex hyperbolic geometry.
Schwarz lemma and equality
Theorem (Schwarz). If \(f:\mathbb D\to\mathbb D\) is holomorphic and \(f(0)=0\), then \[ |f(z)|\le|z|,\qquad |f'(0)|\le1. \] If equality holds at one nonzero point or in the derivative bound, then \(f(z)=e^{i\alpha}z\).
Proof. Define \(g(z)=f(z)/z\) for \(z\ne0\) and \(g(0)=f'(0)\). For \(0<r<1\), the maximum principle on \(|z|\le r\) gives \(|g(z)|\le1/r\); let \(r\uparrow1\). Equality at an interior point makes \(g\) constant of unit modulus.
Automorphisms of the disc
For \(a\in\mathbb D\), \[ \phi_a(z)=\frac{a-z}{1-\bar a z} \] is an involutive automorphism exchanging \(0\) and \(a\). Every automorphism of \(\mathbb D\) is \[ e^{i\theta}\phi_a(z). \] Indeed, compose an automorphism with \(\phi_{f(0)}\) to fix zero and apply Schwarz to the map and its inverse.
Schwarz--Pick and the pseudohyperbolic metric
Theorem (Schwarz--Pick). Every holomorphic \(f:\mathbb D\to\mathbb D\) satisfies \[ \left| \frac{f(z)-f(w)}{1-\overline{f(w)}f(z)} \right| \le \left|\frac{z-w}{1-\bar wz}\right|, \] and \[ \frac{|f'(z)|}{1-|f(z)|^2} \le\frac1{1-|z|^2}. \] Equality at one nontrivial pair or derivative point occurs exactly for disc automorphisms.
Proof. Apply Schwarz's lemma to \[ \phi_{f(w)}\circ f\circ\phi_w, \] which fixes \(0\). The value inequality is Schwarz's bound. Differentiate at zero and use \(|\phi_a'(a)|=1/(1-|a|^2)\) to obtain the infinitesimal inequality. Equality uses the equality case of Schwarz.
The line element \[ \mathrm{d}s_{\mathbb D}=\frac{2|\mathrm{d}z|}{1-|z|^2} \] is therefore contracted by holomorphic self-maps and preserved by automorphisms. Its integrated distance is \[ d_{\mathbb D}(z,w) =2\operatorname{artanh} \left|\frac{z-w}{1-\bar wz}\right|. \]
Denjoy--Wolff behavior
If a holomorphic self-map of the disc is not an elliptic automorphism, its iterates converge locally uniformly to a unique point in \(\overline{\mathbb D}\), the Denjoy--Wolff point. Schwarz--Pick supplies the metric contraction behind this theorem, though boundary fixed-point analysis is needed when the limit lies on \(\partial\mathbb D\).