Maximum modulus principle

Maximum modulus principle. If $f$ is analytic on a bounded domain $\Omega$ and continuous on $\overline{\Omega }$, then $|f|$ attains its maximum on the boundary $\partial \Omega$.

Furthermore, if $|f|$ attains its maximum at an interior point of $\Omega$, then $f$ is constant on $\Omega$.

A non-constant analytic function cannot have an interior maximum of its modulus.

Sketch of proof

Suppose $|f|$ attains an interior maximum at $z_0\in \Omega$. By the Cauchy integral formula applied to a small circle of radius $r$ around $z_0$:

$f(z_0)=\frac{1}{2\pi }{\int }_0^{2\pi }f(z_0+r{e}^{i\theta })d\theta .$

Taking modulus:

$|f(z_0)|\le \frac{1}{2\pi }{\int }_0^{2\pi }|f(z_0+r{e}^{i\theta })|d\theta \le |f(z_0)|$

(the last step uses the maximum assumption: $|f(z)|\le |f(z_0)|$ for $z$ near $z_0$). The squeeze forces $|f(z_0+r{e}^{i\theta })|=|f(z_0)|$ for all $\theta$ — i.e., $|f|$ is constant on the circle of radius $r$.

Since $r>0$ was arbitrary (any small $r$ works), $|f|$ is constant on a whole neighborhood of $z_0$. From $|f|$ locally constant, conclude $f$ locally constant: write $|f{|}^2=f\bar{f}=c$ (a positive real constant) on the neighborhood. If $c=0$ then $f\equiv 0$ on the neighborhood, done. Else differentiate $f\bar{f}=c$ with respect to $x$ and $y$ to get $f_x\bar{f}+f{\bar{f}}_x=0$ and $f_y\bar{f}+f{\bar{f}}_y=0$. Using the Cauchy-Riemann equations on $f$ analytic, $f_y=i{f}_x$, and on $\bar{f}$ analytic-in-$\bar{z}$, ${\bar{f}}_y=-i{\bar{f}}_x$. Substituting and combining the two equations gives $f_x\bar{f}=0$ throughout. Since $\bar{f}\ne 0$, $f_x\equiv 0$ on the neighborhood, so $f$ is constant there.

Analytic continuation (using the identity theorem: analytic functions agreeing on an open set agree everywhere on a connected domain) extends ”$f$ constant on a neighborhood of $z_0$” to ”$f$ constant on all of the connected domain $\Omega$“. ∎

What it means

A non-constant analytic function is “spread out” — its modulus doesn’t have hills with peaks in the interior. The max of $|f|$ always lives on the boundary.

Real functions don’t behave this way. A polynomial $p(x)$ on $[0,1]$ can have any max it wants in the interior — say, $x(1-x)$ peaks at $1/2$. The complex version forbids this.

Minimum modulus principle

A companion: if $f$ is analytic on $\overline{\Omega }$, $f\ne 0$ in $\Omega$, and $f$ is continuous on $\overline{\Omega }$, then $|f|$ attains its minimum on the boundary.

Proof: apply the maximum modulus principle to $1/f$, which is also analytic (since $f$ is nonzero in $\Omega$).

The condition $f\ne 0$ in $\Omega$ is essential. Without it, $|f|$ trivially attains its min ($=0$) in the interior.

Connection to harmonic functions

The same statement holds for harmonic functions: a non-constant harmonic function on a bounded domain attains its max (and min) on the boundary. Since $\log |f|$ is harmonic when $f$ is analytic and nonzero, the maximum modulus principle is essentially the max principle for $\log |f|$.

Harmonic functions inherit this from the mean-value property (value at center of disk = average over bounding circle), which forbids interior maxima — and the mean-value property follows directly from CIF, the same way.

In context

The maximum modulus principle is used to prove:

• Uniqueness in boundary-value problems. Two analytic functions agreeing on the boundary differ by something with zero boundary values; the maximum principle forces them equal in the interior.
• Schwarz lemma: an analytic $f:\text{disk}\to \text{disk}$ with $f(0)=0$ satisfies $|f(z)|\le |z|$. Foundational for conformal mapping.
• Phragmén–Lindelöf principle: refinements for unbounded domains.

It also appears in physics applications: the steady-state potential in a region with no internal sources can’t have an interior maximum. Heat reaches its hot spots at the boundary heat sources, not in the middle of a cool interior.