Maximum Modulus Principle

Nonconstant holomorphic |f| peaks only on the boundary.

The idea

Theorem (Maximum Modulus Principle).

Let $f$ be holomorphic and nonconstant on a domain, that is, a connected open set. Then $|f|$ has no local maximum at any point of the domain. In particular, if $f$ is holomorphic on a bounded domain and continuous on its closure, then $|f|$ attains its maximum on the boundary.

The second form is the one most often used: on a closed bounded region the largest value of $|f|$ is found by searching the boundary alone, since no interior point can be a peak.

Each hypothesis is needed. A constant $f$ has $|f|$ maximal at every point, so nonconstancy cannot be dropped. And without a closed bounded region the maximum may fail to exist: $|e^{z}| = e^{\operatorname{Re} z}$ has no largest value on the open right half-plane.

Ways to work on it

• Walkthrough. Mean value property, no interior maximum, and the boundary consequence.
• Practice. Find the boundary maximum of a holomorphic function on the unit disk.
• Hardest. Prove the Schwarz lemma using the maximum modulus principle.

Not sure where to start? Take the ten-question placement test.