Schwarz's Lemma and Automorphisms of the Disk

Core Idea

Schwarz's lemma says that if f is holomorphic on the open unit disk D, maps D into D, and fixes the origin, then |f(z)| ≤ |z| for all z in D and |f'(0)| ≤ 1. If equality |f(z)| = |z| holds at a single nonzero point, or if |f'(0)| = 1, then f is a rotation, f(z) = e^iθz. A short argument with the maximum modulus principle gives this rigid conclusion. Combined with Möbius transformations, it identifies every conformal self-map of the disk as a map of the form e^iθ(z - a)/(1 - āz) with |a| < 1.

How It's Best Learned

Prove the lemma by applying the maximum modulus principle to g(z) = f(z)/z on circles |z| = r and letting r approach 1. Then check by direct computation that φₐ(z) = (z - a)/(1 - āz) maps the disk to itself and sends a to 0. Use φₐ to move any point to the origin, which is the step that lets the lemma apply to maps that do not fix 0.

Common Misconceptions

• Schwarz's lemma applies to any bounded holomorphic function on the disk. (It needs f(0) = 0 and |f| < 1 on D; for maps not fixing 0, first compose with a disk automorphism.)
• The bound |f'(0)| ≤ 1 means |f'(z)| ≤ 1 everywhere in the disk. (The lemma bounds the derivative only at the origin; f(z) = z² has |f'(z)| = 2|z|, which exceeds 1 for |z| > 1/2.)
• Any one-to-one holomorphic map of the disk onto itself fixing 0 might be complicated. (By the equality case, it must be a rotation z ↦ e^iθz.)

Explainer

Holomorphic functions are surprisingly rigid, and Schwarz's lemma is a sharp example. It concerns a holomorphic map f of the open unit disk D = {|z| < 1} into itself with f(0) = 0.

The lemma. Under these hypotheses, |f(z)| ≤ |z| for every z in D and |f'(0)| ≤ 1. Moreover, if |f(z₀)| = |z₀| for some z₀ ≠ 0, or if |f'(0)| = 1, then f(z) = e^iθz for some real θ.

The proof is short. Since f(0) = 0, the power series of f has no constant term, so g(z) = f(z)/z extends to a holomorphic function on D with g(0) = f'(0). Fix r < 1. On the circle |z| = r we have |g(z)| = |f(z)|/r < 1/r. By the maximum modulus principle, |g(z)| ≤ 1/r on the closed disk |z| ≤ r. Letting r increase to 1 gives |g(z)| ≤ 1 on all of D, which is |f(z)| ≤ |z|, and at z = 0 it gives |f'(0)| ≤ 1. If |g| equals 1 at an interior point, the maximum modulus principle forces g to be a constant of modulus 1, so f is a rotation.

The lemma has a geometric reading: a holomorphic self-map of the disk that fixes the center cannot push any point farther from the center. It can only rotate the disk or pull points inward.

The condition f(0) = 0 is removed with a family of Möbius transformations. For |a| < 1, let

φₐ(z) = (z - a)/(1 - āz).

A direct calculation shows |φₐ(z)| < 1 when |z| < 1, that φₐ(a) = 0, and that its inverse is φ₋ₐ. So φₐ is a holomorphic bijection of the disk onto itself, an automorphism, that moves a to the origin.

These maps describe every automorphism. Suppose h is a holomorphic bijection of D onto D with h(a) = 0. Then F = h ∘ φ₋ₐ is an automorphism fixing 0. Applying Schwarz's lemma to F and to its inverse gives |F(z)| ≤ |z| and |z| ≤ |F(z)|, so |F(z)| = |z| and F is a rotation. Therefore h(z) = e^iθ(z - a)/(1 - āz). The automorphisms of the disk are exactly these maps.

This classification is a key step in the proof of the Riemann mapping theorem, where it shows that the conformal map onto the disk is unique once a point and a direction are fixed.