Maximum Modulus Principle for Polynomials

The maximum modulus principle states that a non-constant polynomial cannot attain its maximum modulus in the interior of a region. This is a precursor to the general maximum modulus principle for analytic functions.

Statement

Theorem1.4Maximum modulus for polynomials

Let $p(z)$ be a non-constant polynomial and let $D$ be a bounded domain in $\mathbb{C}$ . Then the maximum of $|p(z)|$ on $\overline{D}$ (the closure of $D$ ) is attained on the boundary $\partial D$ , never in the interior.

RemarkInterpretation

This means: if $z_0 \in D$ (interior), then there exists $z_1 \in D$ with $|p(z_1)| > |p(z_0)|$ . The function $|p(z)|$ has no local maxima in the interior — any maximum must occur on the boundary.

Proof Idea

Proof

Suppose for contradiction that $|p(z)|$ attains its maximum at some $z_0 \in D$ (interior point). Then for all $z$ in a small disk around $z_0$ , we have $|p(z)| \leq |p(z_0)|$ .

Write $p(z) = p(z_0) + (z - z_0)^k q(z)$ where $k \geq 1$ is the order of the first nonzero derivative at $z_0$ and $q(z_0) \neq 0$ . Choose $\theta$ such that $e^{ik\theta} q(z_0) / |q(z_0)| = 1$ (i.e., $e^{ik\theta} q(z_0)$ points in the direction of $p(z_0)$ ).

For small $\epsilon > 0$ , let $z = z_0 + \epsilon e^{i\theta}$ . Then

$p(z) = p(z_0) + \epsilon^k e^{ik\theta} q(z_0) + O(\epsilon^{k+1}).$

Choosing $\theta$ appropriately, we can make $|p(z)| > |p(z_0)|$ , contradicting the assumption that $|p(z_0)|$ is a maximum.

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Examples

ExampleLinear polynomial

$p(z) = z$ on the unit disk $|z| \leq 1$ . The maximum of $|p(z)| = |z|$ is $1$ , attained at all points on the boundary circle $|z| = 1$ , never in the interior.

ExampleQuadratic polynomial

$p(z) = z^2$ on the closed disk $|z| \leq 2$ . The maximum of $|p(z)| = |z|^2$ is $4$ , attained on the boundary $|z| = 2$ . At the origin (interior), $|p(0)| = 0 < 4$ .

ExampleNon-constant on a square

$p(z) = z^3 - z$ on the square $[-1, 1] \times [-1, 1]$ . The maximum of $|p(z)|$ must occur on the boundary of the square (the four edges), not at any interior point like $(0, 0)$ or $(0.5, 0.5)$ .

Generalizations

RemarkMaximum modulus for analytic functions

The maximum modulus principle extends to all analytic functions, not just polynomials. If $f$ is analytic and non-constant on a domain $D$ , then $|f(z)|$ has no local maxima in $D$ . This is a fundamental theorem in complex analysis, proved using the Cauchy integral formula.

See Maximum modulus principle for the general version.

RemarkMinimum modulus

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

Applications

ExampleUniqueness on the boundary

If two polynomials $p$ and $q$ agree on the boundary of a domain $D$ , then $|p - q|$ attains its maximum on the boundary. If $p(z) = q(z)$ for all $z \in \partial D$ , then $|p(z) - q(z)| = 0$ everywhere on $\partial D$ , so $|p - q|$ is identically zero in $\overline{D}$ (by continuity and the maximum modulus principle). Thus $p = q$ on $\overline{D}$ .

This is a uniqueness principle: a polynomial is determined by its boundary values.

ExampleApproximation bounds

If $|p(z)| \leq M$ on the boundary $\partial D$ , then $|p(z)| \leq M$ for all $z \in D$ . This is useful for bounding polynomials: to control $|p|$ everywhere, it suffices to control it on the boundary.

Summary

RemarkKey takeaways

A non-constant polynomial $p(z)$ attains its maximum modulus on the boundary, never in the interior.
This is a special case of the maximum modulus principle for analytic functions.
The principle implies uniqueness: boundary values determine the function.
The minimum modulus principle does not hold in general (unless the function is nonvanishing).
The maximum modulus principle is one of the most important results in complex analysis, with applications to uniqueness, approximation, and the theory of harmonic functions.