Maximum Modulus Principle

The maximum modulus principle is one of the most important theorems in complex analysis. It states that a non-constant holomorphic function cannot attain its maximum modulus in the interior of its domain.

Statement

Theorem2.2Maximum modulus principle

Let $f$ be holomorphic on a domain $D \subseteq \mathbb{C}$ and continuous on $\overline{D}$ (the closure). If $f$ is non-constant, then $|f(z)|$ attains its maximum on the boundary $\partial D$ , never in the interior.

More precisely: if $z_0 \in D$ (interior) and $|f(z_0)| \geq |f(z)|$ for all $z$ in a neighborhood of $z_0$ , then $f$ is constant.

RemarkContrapositive form

If $|f|$ has a local maximum at an interior point, then $f$ is constant on the entire connected component containing that point.

Proof Sketch

RemarkProof via Cauchy integral formula

Suppose $|f(z_0)| \geq |f(z)|$ for all $z$ in a disk $|z - z_0| < r$ . By Cauchy's integral formula,

$f(z_0) = \frac{1}{2\pi i} \oint_{|z-z_0|=r} \frac{f(z)}{z - z_0} dz = \frac{1}{2\pi} \int_0^{2\pi} f(z_0 + re^{i\theta}) d\theta.$

This is the mean value property: $f(z_0)$ equals the average of $f$ on the circle. Taking moduli:

$|f(z_0)| \leq \frac{1}{2\pi} \int_0^{2\pi} |f(z_0 + re^{i\theta})| d\theta \leq \max_{|z-z_0|=r} |f(z)|.$

If $|f(z_0)|$ is maximal, then $|f(z)| = |f(z_0)|$ on the entire circle. By the strict triangle inequality, this forces $f(z)$ to be constant on the circle. Repeating this argument shows $f$ is constant in the disk, and by analytic continuation, constant on $D$ .

Examples and Applications

ExampleMaximum of a polynomial

Let $p(z) = z^3$ on the disk $|z| \leq 2$ . Then $|p(z)| = |z|^3 \leq 8$ , with equality only on the boundary $|z| = 2$ . The maximum is $8$ , attained at all points on $|z| = 2$ .

ExampleExponential function

On the rectangle $[-1, 1] \times [-1, 1]$ , the maximum of $|e^z| = e^{\text{Re}(z)}$ occurs at $z = 1 + iy$ (the right edge), giving $|e^z| = e$ . The minimum occurs at $z = -1 + iy$ (the left edge), giving $|e^z| = e^{-1}$ .

ExampleNon-constant implies no interior max

If $f(z) = z$ on the unit disk, then $|f(z)| = |z| < 1$ for all interior points, and $|f(z)| = 1$ only on the boundary. The maximum modulus principle holds.

Minimum Modulus Principle

RemarkNo minimum modulus principle in general

There is no minimum modulus principle: $|f(z)|$ can attain its minimum in the interior (e.g., at zeros). However, if $f$ is non-zero on $D$ , then $1/f$ is holomorphic, and applying the maximum modulus principle to $1/f$ gives a minimum modulus principle for nonvanishing functions.

Theorem2.3Minimum modulus principle (nonvanishing case)

If $f$ is holomorphic and non-zero on a domain $D$ , and continuous on $\overline{D}$ , then $|f(z)|$ attains its minimum on the boundary $\partial D$ .

Applications

ExampleUniqueness from boundary data

If two holomorphic functions $f$ and $g$ on $D$ agree on $\partial D$ , then $|f - g|$ attains its maximum on $\partial D$ . Since $|f(z) - g(z)| = 0$ on $\partial D$ , we have $|f - g| \equiv 0$ on $\overline{D}$ , so $f = g$ on $D$ .

This is a uniqueness principle: boundary values determine the function.

ExampleSchwarz lemma

The maximum modulus principle is used to prove the Schwarz lemma: if $f: D(0,1) \to D(0,1)$ is holomorphic with $f(0) = 0$ , then $|f(z)| \leq |z|$ and $|f'(0)| \leq 1$ . See Schwarz lemma.

ExamplePhragmén-Lindelöf principle

The maximum modulus principle extends to unbounded domains under growth conditions. For instance, if $f$ is holomorphic in a sector and $|f(z)|$ is bounded on the boundary rays and grows at most like $e^{|z|^\alpha}$ for some $\alpha < 1$ , then $|f|$ is bounded in the sector.

Summary

RemarkKey takeaways

A non-constant holomorphic function attains its maximum modulus on the boundary, never in the interior.
This is a consequence of the mean value property (from Cauchy's integral formula).
The minimum modulus principle does not hold unless the function is nonvanishing.
Applications: uniqueness, Schwarz lemma, Phragmén-Lindelöf, and many others.
The maximum modulus principle is one of the most powerful tools in complex analysis, with applications to PDE, approximation theory, and potential theory.