The Schwarz Lemma

The setup: self-maps of the disk that fix the centre

You arrive at this rung already fluent with the unit disk as the friendly region every conformal problem wants to be transplanted to, and with the maximum-modulus principle as the rigid law that a non-constant holomorphic function attains its largest |f| only on the boundary. The Schwarz lemma is what you get when you point that law at the most basic situation imaginable: a map that takes the disk into itself and leaves the centre alone. The astonishing thing is how much such an innocent setup is forced to obey.

Let me fix notation we will keep all rung long. Write D for the open unit disk, the set of all z with |z| < 1. Suppose f is a holomorphic function on D whose values never leave the disk, so |f(z)| < 1 for every z in D, and which pins the centre, f(0) = 0. That is the entire hypothesis: holomorphic, disk into disk, origin fixed. No smoothness assumed at the boundary, no formula for f, nothing else.

From that bare hypothesis the Schwarz lemma squeezes out two crisp conclusions. First, f does not push any point farther from the centre than the point already was: |f(z)| is at most |z| for every z in D. Second, the map cannot stretch at the centre beyond rate one: the derivative satisfies |f'(0)| is at most 1. Both are upper bounds with the SAME constant — the identity map z gives equality in both — and the rest of the guide is about why that bound holds and what happens at the knife-edge when it is attained.

The trick: divide by z and let the maximum principle work

The proof is one of those moves that looks like a magic trick until you see it is forced. The hypothesis f(0) = 0 says z divides f, so form the quotient g(z) = f(z) / z. At every non-zero point of D this is an honest holomorphic function. The only worry is z = 0, where we seem to be dividing zero by zero — but that is precisely a removable singularity, and filling the hole with the limit value gives g(0) = f'(0). So g is holomorphic on the whole disk, with no exception at the centre.

g(z) = f(z) / z          for z =/= 0
g(0) = f'(0)             (the removable-singularity value)

On the circle |z| = r:   |g(z)| = |f(z)| / r  <  1 / r
                                              (since |f| < 1)

Max-modulus on the disk |z| <= r:
        |g(z)| <= max over |z| = r  <  1 / r   for ALL |z| <= r

Let r -> 1 :              |g(z)| <= 1   throughout D

Here is the engine. On the circle of radius r (with r < 1) we have |g(z)| = |f(z)| / r, and since |f| < 1 this is strictly less than 1/r. The maximum-modulus principle now insists that |g| inside the disk of radius r can be no larger than its biggest value on that boundary circle, so |g(z)| < 1/r everywhere with |z| at most r. That inequality holds for every r below 1, and we are free to let r climb toward 1. As it does, 1/r relaxes down to 1, leaving the clean conclusion |g(z)| is at most 1 throughout D.

Unfold the definition of g and both headline claims fall out at once. From |g(z)| at most 1 we read |f(z)| at most |z|, the first conclusion, valid for every z in D. And evaluating at the centre, |g(0)| = |f'(0)| is at most 1, the second. Notice how little we used: only that f maps the disk into the disk, that it fixes the origin, and the single estimate the maximum-modulus principle hands us. That economy is the whole charm of the lemma.

The rigid case: equality forces a rotation

So far we have two inequalities, and inequalities feel soft. The real teeth of the Schwarz lemma are in its equality clause, which is anything but soft. Suppose the bound is touched even once: either |f(z_0)| = |z_0| at a single non-zero point z_0, OR |f'(0)| = 1 at the centre. Then f is not merely close to the identity-sized limit — it is forced to be exactly a rotation, f(z) = e^(i theta) z for one fixed real angle theta. One point of equality, and the entire function is pinned down.

The reason reuses the very same g, now caught at its own ceiling. Equality means |g| reaches 1 somewhere INSIDE the open disk — at z_0 in the first case, at the origin in the second. But the maximum-modulus principle says a holomorphic function whose modulus attains an interior maximum must be constant. So g is a constant of modulus 1, that is g(z) = e^(i theta) for some real theta. Multiply back by z and you have f(z) = e^(i theta) z exactly: a pure rotation, no stretching, no bending.

What it really says, and what it does not

It is worth saying in plain words what the lemma asserts, because the slogan 'maps shrink toward the centre' is a half-truth. The honest reading is comparative: the identity is the most expansive self-map of the disk that fixes the origin, and every other such map is, at the centre and along the way out, no more expansive than it. A map can certainly move some individual points farther from where simpler maps would, but it can never beat the identity's outward reach |f(z)| at most |z|, and it can never out-stretch the identity at the centre. The identity sits at the extreme, and Schwarz says nothing else gets past it.

Be equally clear about the load-bearing hypotheses, because dropping any one breaks the conclusion. The fixed-centre condition f(0) = 0 is essential — without it the quotient f(z)/z need not be holomorphic, and indeed a map that slides the disk sideways can push the origin's image outward, so |f(z)| at most |z| simply fails. The into-the-disk condition |f| < 1 is what makes |g| < 1/r on each circle; relax it and the engine stalls. And holomorphy is non-negotiable: a merely real-differentiable map of the disk obeys none of this, since the maximum-modulus principle, the heart of the proof, is a strictly complex-analytic fact.

Why such a small lemma opens a whole rung

A natural objection is that the hypothesis f(0) = 0 feels special — most self-maps of the disk do not fix the centre, so what use is a lemma that demands it? The answer, which the next guide develops in full, is that the disk has a rich supply of its own symmetries, the disk automorphisms, holomorphic bijections of D onto itself. Each is built from a Blaschke factor of the form (z - a) / (1 - a-bar z), and these can slide any chosen point to the centre. So to study a map that fixes some other point, you pre- and post-compose with automorphisms to drag that point to the origin, apply Schwarz there, and translate the result back.

That conjugation trick is exactly how the Schwarz-Pick lemma, two guides ahead, upgrades today's centred statement into one valid at every point of the disk at once. And the upgrade has a startling reading: the quantity Schwarz controls turns out to be a genuine notion of distance — the hyperbolic metric — under which holomorphic self-maps never increase distances, and the disk automorphisms preserve them exactly. So this featherweight lemma is really the doorway to seeing the disk as a model of non-Euclidean geometry, which is where this rung is headed.