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The Maximum Modulus Principle

A holomorphic function can never have a peak in |f| hidden inside its domain — the biggest value of |f| always lives on the boundary. We trace this to the averaging built into Cauchy's formula, see why a flat spot forces the function to be constant, and watch it power a strikingly clean proof of the Schwarz lemma.

No peaks inside

Picture the modulus |f(z)| of a holomorphic function as the height of a landscape over the complex plane — at each point z you stand and read off how big f is there. Your everyday intuition, sharpened on smooth real surfaces, expects such a landscape to have hills: local summits sitting peacefully somewhere in the interior, surrounded on all sides by lower ground. The maximum modulus principle says, flatly and without exception, that no such interior summit can exist. If f is holomorphic on a domain and not constant, then |f| has no local maximum anywhere inside. Every peak you might hope to find has been pushed out to the edge.

The clean working form is about a bounded region together with its boundary. Suppose f is holomorphic on a bounded domain D and continuous up to the boundary. Then the largest value of the modulus |f| over the closed region is attained on the boundary curve — and, unless f is constant, it is attained ONLY there. So to find the maximum of |f| over a disk, a square, or any nice region, you never have to search the interior: you walk the boundary and look there. The whole inside is, in a sense, the shadow of its rim.

Pause on how alien this is to real-variable life. A real smooth function like the bump u(x) = 1 - x^2 has a glorious interior maximum at x = 0; that is the most ordinary thing in the world. The reason |f| cannot behave this way is not that complex functions are tame — it is that holomorphy is an enormously rigid condition, the same rigidity that earlier in this rung forced one complex derivative to deliver infinitely many. Here that rigidity expresses itself as a refusal to bulge.

Why averaging forbids a summit

The engine is a fact you already met in the opening guide of this rung. Cauchy's formula on a circle, with z_0 at the centre, says f(z_0) is the AVERAGE of the boundary values of f around that circle — this is the mean value property. Setting z = z_0 + r e^(i theta), the formula collapses to f(z_0) = (1/2 pi) times the integral of f(z_0 + r e^(i theta)) over theta from 0 to 2 pi. The value at the centre is not bigger than its neighbours; it is exactly their mean.

Mean value property (Cauchy's formula on a circle of radius r):

  f(z_0) = (1/2 pi) * integral_0^{2 pi} f(z_0 + r e^(i theta)) d theta

Take modulus and pull it inside the average:

  |f(z_0)| <= (1/2 pi) * integral_0^{2 pi} |f(z_0 + r e^(i theta))| d theta

So |f| at the centre <= the average of |f| around the circle.
The centre value is the average of the boundary values — and an average can never exceed the things it averages.

Now squeeze. Take the modulus of both sides and use the triangle inequality (the modulus of an average is at most the average of the moduli): |f(z_0)| is at most the average of |f| around the circle. Suppose, for contradiction, that z_0 were a local maximum of |f|. Then on a small enough circle every boundary value |f| would be at most |f(z_0)|. But the centre equals their average, and an average of numbers that are all <= |f(z_0)| can equal |f(z_0)| only if every one of them equals |f(z_0)|. So |f| would have to be constant on that whole circle — and then, sweeping over all small radii, constant on a whole little disk.

The last step seals it. A holomorphic function whose modulus |f| is constant on an open disk must itself be constant there (a quick Cauchy-Riemann calculation: if |f|^2 = u^2 + v^2 is constant and f is holomorphic, f' turns out to be 0). And a holomorphic function that is constant on a small disk is constant on the entire connected domain, by the identity theorem from the holomorphic rung. So the only way to have an interior maximum is to be constant everywhere — which is exactly the principle, stated as a contradiction.

The minimum, and where it can hide

It is natural to ask the mirror question: does |f| also refuse to have an interior minimum? Mostly yes — but with one honest exception you must respect. If f has no zero inside the domain, then 1/f is also holomorphic, and applying the maximum principle to 1/f turns a minimum of |f| into a maximum of |1/f|, which is forbidden in the interior. This is the minimum modulus principle: a non-vanishing, non-constant holomorphic function attains its smallest |f| on the boundary too.

There is a deeper reading of all this, and it connects to harmonic functions from earlier on the ladder. Writing f = u + i v, the real part u is a harmonic function, and harmonic functions obey their own maximum principle: u, too, takes its largest and smallest values on the boundary. That is no coincidence. The modulus version and the harmonic version are two faces of the same averaging fact — the value at a centre is the mean of the values around it — which is precisely why the whole circle of ideas radiates out from Cauchy's formula.

An open door: why |f| has no peak

There is a second, more geometric way to see the principle, and it is worth knowing because it reveals what is really going on. A non-constant holomorphic function is an open map: it sends open sets to open sets. This is the open mapping theorem. The intuition is that near a point where f' is not zero, f acts like a tiny rotation-and-scaling — an amplitwist — so a small disk around z_0 maps onto a small (slightly tilted, slightly resized) disk around f(z_0), with f(z_0) sitting in its interior, not on its rim.

Now the punchline is immediate. If |f| had a maximum at z_0, then f(z_0) would be a point of the image as far from the origin as possible — a point on the OUTER edge of the image set, with nothing of the image beyond it. But the open mapping theorem says f(z_0) is an interior point of the image, so there are image points strictly farther out, hence points with strictly larger |f| nearby. That contradicts z_0 being a maximum. The image is not allowed to have an edge that the interior reaches, and |f| measures distance to that edge.

Notice how the maximum modulus principle now sits inside a small constellation of consequences that all flow from Cauchy's formula in this rung: infinite differentiability, the Cauchy estimates, Liouville's theorem and through it the fundamental theorem of algebra, Morera's converse, and the open mapping theorem. They are not a grab-bag; they are different pressure points of the one rigidity, and the maximum principle is the one you can almost SEE in the landscape picture.

Putting it to work: the Schwarz lemma

The maximum principle is not just scenery; it is a sharp tool for pinning functions down. The cleanest demonstration is the Schwarz lemma, a small result with an outsized influence on conformal geometry. The setup: f maps the open unit disk into itself, is holomorphic, and fixes the centre, f(0) = 0. The conclusion is strikingly tight: |f(z)| <= |z| everywhere, and |f'(0)| <= 1 — a holomorphic self-map that pins the origin cannot expand anything.

  1. Since f(0) = 0, the quotient g(z) = f(z)/z has a removable singularity at 0 and extends to a holomorphic function on the whole disk (this uses the power-series fact from this rung: f starts with a z term, so dividing by z is clean).
  2. On the circle |z| = r (just inside the disk), |f| < 1, so |g(z)| = |f(z)|/r < 1/r. By the maximum modulus principle, that bound on the boundary circle holds throughout the disk |z| <= r as well.
  3. Now let r climb to 1. The bound |g(z)| <= 1/r tightens to |g(z)| <= 1 at every fixed point of the disk. That says |f(z)| <= |z|; evaluating g at 0 gives g(0) = f'(0), so |f'(0)| <= 1 as well.
  4. The rigidity clause: if equality |f(z)| = |z| holds at even ONE interior point, then |g| reaches its maximum 1 inside, so by the principle g is a constant of modulus 1, forcing f(z) = (constant) times z — a pure rotation.