the Schwarz lemma
/ shvarts /
Imagine you live inside the unit disk — the set of all complex numbers z with |z| < 1 — and you are only allowed to move points around using a holomorphic map that keeps you inside the disk and pins the center 0 in place. The Schwarz lemma says you have far less freedom than you might expect. You cannot push any point farther out from the center than where it started, and you cannot stretch things at the center either. The disk, under holomorphic self-maps fixing the origin, is rigid: it resists being expanded from within.
Precisely: let f be holomorphic on the open unit disk, with |f(z)| < 1 for all z there, and f(0) = 0. Then two inequalities hold for every z: |f(z)| <= |z|, and at the center |f'(0)| <= 1. Moreover the inequalities are sharp in a striking way — if equality |f(z_0)| = |z_0| happens at even a single point z_0 other than 0, or if |f'(0)| = 1, then f must be an exact rotation, f(z) = e^(i theta) z for some fixed angle theta. The proof is a clean trick: the function g(z) = f(z)/z has a removable singularity at 0 (because f(0) = 0), so g is holomorphic on the disk; on the circle |z| = r < 1 we have |g(z)| = |f(z)|/r < 1/r, and letting r -> 1 the maximum modulus principle forces |g(z)| <= 1 everywhere, which is exactly |f(z)| <= |z|. If |g| reaches 1 inside, the maximum principle says g is a constant of modulus 1 — a rotation.
This little lemma is one of the most powerful rigidity statements in complex analysis. It is the engine behind classifying all the conformal automorphisms of the disk, behind the Schwarz-Pick lemma and hyperbolic geometry, and behind countless uniqueness and bounding arguments. A common surprise: on the real line, a smooth map of an interval into itself fixing a point can do almost anything near that point, but in the complex world the single condition f(0) = 0 plus 'stays in the disk' clamps the whole map down to |f(z)| <= |z|. That is the magic of holomorphy — one complex constraint propagates globally.
Take f(z) = z^2 on the disk. It is holomorphic, maps the disk into itself, and f(0) = 0, so the lemma applies: indeed |z^2| = |z|^2 <= |z| since |z| < 1, and f'(0) = 0, comfortably below 1. By contrast f(z) = e^(i theta) z is the boundary case — it achieves |f(z)| = |z| everywhere and |f'(0)| = 1, and it is a pure rotation, exactly as the equality clause predicts.
z^2 obeys the strict inequalities; a rotation e^(i theta) z is the rigid equality case.
The two hypotheses are both essential: drop f(0) = 0 and the bound |f(z)| <= |z| simply fails (you need the Schwarz-Pick version). The lemma is stated for the unit disk and origin specifically; for other centers, compose with a disk automorphism first.