Maximum Modulus Principle — Mathematics Knowledge System

Overview

The maximum modulus principle says that if f is holomorphic and non-constant on a connected open domain, then |f| cannot attain a local maximum at any interior point of that domain — every local max of |f| must occur on the boundary. Equivalently, on a compact domain, the maximum of |f| over the closed region is attained on the boundary (never strictly inside, unless f is constant). This is a direct consequence of the mean value property of holomorphic functions: since f(z₀) is the average of f over any small circle around z₀, |f(z₀)| cannot exceed the maximum of |f| on that circle unless f is constant near z₀. The principle is one of the sharpest rigidity results in complex analysis — it rules out an entire class of behaviors (interior 'bumps' in |f|) that are common for smooth real functions, and it underlies uniqueness results, growth estimates like the Schwarz lemma, and many applications in engineering stability analysis.

Intuition

The mean value property says f(z₀) equals the average of f around any small circle centered at z₀. Averaging can never produce a value with strictly larger modulus than every value being averaged — a plain average is always somewhere between the extremes, in modulus. So if |f(z₀)| were a strict local maximum, it would have to exceed the average of |f| on a tiny surrounding circle, which is impossible unless f is literally constant on that circle (and then, by analytic continuation, constant on the whole connected domain). This is exactly what real harmonic functions (which are, not coincidentally, real/imaginary parts of holomorphic functions) also obey: no interior bumps, because interior values are always averages of surrounding values. Physically, this says a non-constant analytic function can't have a local 'hot spot' of magnitude tucked away in the interior — all the extreme behavior gets pushed to the boundary.

Formal Definition

Definition

Let f be holomorphic and non-constant on a domain (open, connected set) Ω:

\text{If } |f| \text{ attains a local maximum at } z_0 \in \Omega, \text{ then } f \text{ is constant on } \Omega

Maximum modulus principle (open form)

\max_{\overline{\Omega}} |f| = \max_{\partial\Omega} |f| \qquad (\Omega \text{ bounded, } f \text{ continuous on } \overline{\Omega})

Boundary maximum form (for bounded domains)

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

Mean value inequality driving the proof

Worked Examples

f is entire (holomorphic everywhere) and non-constant, so by the maximum modulus principle the max of |f| on the closed disk occurs on the boundary |z|=1.
$\max_{|z|\le 1}|f(z)| = \max_{|z|=1}|f(z)|$
On |z|=1, write z = e^{iθ}, so z² = e^{2iθ}, and f(z) = e^{2iθ}+1.
$|f(e^{i\theta})| = |e^{2i\theta}+1|$
|e^{2iθ}+1|² = (cos2θ+1)² + sin²2θ = cos²2θ+2cos2θ+1+sin²2θ = 2+2cos2θ, maximized when cos2θ=1 (θ=0), giving |f|²=4.
$|e^{2i\theta}+1|^2 = 2+2\cos 2\theta \le 4 \;\Rightarrow\; |f| \le 2$

Answer: The maximum of |f| on |z|≤1 is 2, attained at z = 1 (on the boundary), consistent with the maximum modulus principle.

Practice Problems

Difficulty 5/10

Find the maximum of |f(z)| for f(z) = z on the closed disk |z| ≤ 3.

Difficulty 6/10

Find the maximum of |e^z| on the closed rectangle 0 ≤ x ≤ 1, 0 ≤ y ≤ π (z = x+iy).

Difficulty 7/10

A non-constant holomorphic function f satisfies |f(z)| = 5 for every z on |z|=2, and f is holomorphic inside |z|<2. What can you conclude about |f(z)| for |z| < 2?

Quiz

Summary

If f is holomorphic and non-constant on a connected domain, |f| cannot attain a local maximum at any interior point.
On a bounded domain with f continuous up to the boundary, the maximum of |f| over the closed region is attained on the boundary.
The principle follows from the mean value property: f(z₀) is an average of surrounding values, so |f(z₀)| can't strictly exceed them all unless f is constant.
The maximum modulus principle underlies uniqueness arguments, growth bounds like the Schwarz lemma, and stability results in applications.

References

WebsiteWikipedia — Maximum modulus principle