Functions of a complex variable Complex integration Theorem

Maximum modulus principle

Builds on Holomorphic function Domain in the complex plane Mean value property Identity theorem

Statement

Let $D \subseteq \mathbb{C}$ be a domain and let $f : D \to \mathbb{C}$ be holomorphic on $D$ . Assume that $|f|$ has a local maximum at a point $z_0 \in D$ : there is $r > 0$ such that $|f(z)| \le |f(z_0)|$ for each $z$ with $|z - z_0| < r$ . Then $f$ is constant on $D$ .

Remarks

Proof idea: the mean value formula $f(z_0) = \frac{1}{2\pi} \int_0^{2\pi} f(z_0 + \rho e^{i\theta}) \, d\theta$ for small $\rho$ forces $|f| = |f(z_0)|$ near $z_0$ . Then $f$ is constant near $z_0$ , and on $D$ by the identity theorem. Corollary: if $D$ is bounded and $f$ is continuous on the closure $\overline{D}$ and holomorphic on $D$ , then $\max_{\overline{D}} |f|$ is attained at a point of the boundary of $D$ . The minimum of $|f|$ can be inside $D$ , but only at a zero of $f$ . For example, $|z|$ has its minimum on the unit disc at $0$ .

Reading path

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Complex numbers object
Complex plane concept
Real numbers object
Set bounded above definition
Complex exponential definition
Contour integral definition
Power series in the complex plane definition
Real and imaginary parts definition
Supremum definition
Circle integral definition
Disc of convergence definition
Completeness axiom statement
Existence of square roots of nonnegative reals statement
Square root definition
Modulus of a complex number definition
Complex differentiability definition
Domain in the complex plane definition
Holomorphic function definition
Cauchy–Riemann equations statement
Cauchy's integral formula statement
Zero of a holomorphic function definition
Power series are holomorphic in the disc of convergence statement
Holomorphic functions are analytic statement
Holomorphic functions are infinitely differentiable statement
Order of a zero definition
The real part of a holomorphic function is harmonic statement
Isolated zeros theorem statement
Mean value property statement
Identity theorem statement
Maximum modulus principle statement

Where this leads

19 pages build on this one.

Tag	`048F2P90WR`
URI	`m2://048F2P90WR`
Aliases	`maximum-modulus-theorem`
Kind	statement, theorem
Writing	W1 stated
MSC	30C80
Status	This copy of the site has no build data. Run `m2 status 048F2P90WR`.
Uses	Holomorphic function, Domain in the complex plane, Mean value property, Identity theorem
Same as	Wikidata Q1050230 Wikipedia Maximum modulus principle

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