Schwarz Lemma: Statement, Proof and Applications Guide

The Schwarz lemma is a rigidity statement about bounded analytic functions: if f is analytic on the unit disc, fixes the origin (f(0) = 0), and never exceeds 1 in modulus (|f(z)| ≤ 1), then |f(z)| ≤ |z| for every z in the disc, and |f′(0)| ≤ 1. In words, such a function cannot stretch the disc more than the identity map does.

This matters because it is the seed of far-reaching rigidity: it classifies all conformal automorphisms of the disc, underlies the Schwarz–Pick lemma of hyperbolic geometry, and gives sharp derivative bounds used throughout geometric function theory.

Key Concepts

• Unit disc: 𝔻 = {z : |z| < 1}, the natural domain of the lemma.
• Normalisation: the hypotheses f(0) = 0 and |f| ≤ 1 put every bounded analytic map on the same footing via disc automorphisms.
• Extremal case: if |f(z₀)| = |z₀| for some z₀ ≠ 0, or |f′(0)| = 1, then f(z) = e^{iθ}z for some real θ — a pure rotation.
• Disc automorphisms: φₐ(z) = (z − a)/(1 − āz) (times a rotation) are the bijective analytic self-maps of 𝔻.
• Schwarz–Pick lemma: the invariant form: |f′(z)| ≤ (1 − |f(z)|²)/(1 − |z|²) for any analytic f: 𝔻 → 𝔻.

Formulas

• Schwarz lemma: f analytic on 𝔻, f(0) = 0, |f| ≤ 1 ⟹ |f(z)| ≤ |z| and |f′(0)| ≤ 1.
• Proof sketch: g(z) = f(z)/z extends analytically to 0 with g(0) = f′(0). On |z| ≤ r < 1, max |g| ≤ 1/r on |z| = r by the maximum modulus principle; letting r → 1 gives |g(z)| ≤ 1, i.e. |f(z)| ≤ |z| and |f′(0)| ≤ 1.
• Equality case: equality at any non-zero point forces |g| ≡ 1, so g is constant of modulus 1 and f(z) = e^{iθ}z.
• Schwarz–Pick: |(f(z) − f(w))/(1 − f(z)¯f(w)¯)| ≤ |(z − w)/(1 − z̄w)| — analytic maps contract the pseudo-hyperbolic metric.

Worked Example

Verify the Schwarz lemma for f(z) = z²/2 on the unit disc.

1. f is analytic on 𝔻 (a polynomial), f(0) = 0 ✓.
2. For |z| < 1: |f(z)| = |z|²/2 < 1/2 ≤ 1 ✓, so the hypotheses hold.
3. The lemma predicts |f(z)| ≤ |z|: indeed |z|²/2 ≤ |z| ⟺ |z| ≤ 2, true on 𝔻.
4. It predicts |f′(0)| ≤ 1: f′(z) = z, so f′(0) = 0 ≤ 1 ✓.
5. Numeric check at z = 0.6: |f| = 0.36/2 = 0.18 ≤ 0.6 ✓, with strict inequality since f is not a rotation.

Common Misconceptions

• "Equality |f(z)| = |z| at one point is a coincidence." No — it forces f to be a rotation everywhere. That rigidity is the lemma's sharpest content.
• "The lemma applies without f(0) = 0." Not directly; the standard trick is to compose with disc automorphisms sending f(0) to 0, which yields the Schwarz–Pick form.
• "|f′(0)| ≤ 1 means the derivative is bounded by 1 everywhere." The bound is only at the origin; the Schwarz–Pick lemma gives the correct pointwise bound elsewhere.

Real-World Applications

• Automorphism classification: the lemma proves every bijective analytic self-map of 𝔻 is a Möbius map e^{iθ}(z − a)/(1 − āz).
• Riemann mapping uniqueness: the normalisation f(0) = 0, f′(0) > 0 makes the Riemann map unique — proved via Schwarz.
• Complex dynamics: contraction properties of analytic maps on the disc control iteration behaviour near attracting fixed points.

The Schwarz lemma says a bounded analytic map fixing the origin is dominated by the identity: |f(z)| ≤ |z| and |f′(0)| ≤ 1, with equality only for rotations. The proof is a one-line application of the maximum modulus principle to f(z)/z — yet it unlocks the classification of disc automorphisms and the whole Schwarz–Pick machinery.