Maximum modulus principle on arvau

Wikipedia / Maximum modulus principle

In mathematics, the maximum modulus principle in complex analysis is a holomorphic function, then the modulus cannot exhibit a strict maximum that is strictly within the domain.

is locally a constant function takes larger values.

Formal statement

be a holomorphic function on some bounded and connected open subset of the complex plane such that

in some neighborhood.

This statement can be viewed as a special case of the open mapping theorem is constant.

Related statement

is compact also attains the same maximum at any point of the boundary.

Minimum modulus principle

such that

in some neighborhood.

Sketches of proofs

Using the maximum principle for harmonic functions

One can use the equality

for complex natural logarithms is a harmonic function is a local maximum for this function also, it follows from the maximum principle is constant. Then, using the Cauchy–Riemann equations.

Using Gauss's mean value theorem

Another proof works by using Gauss's mean value theorem is constant.

Using Cauchy's Integral Formula

Source:

. Invoking Cauchy's integral formula, we obtain

Physical interpretation

A physical interpretation of this principle comes from the heat equation, the heat at this maximum would be dispersing to the points around it, which would contradict the assumption that this represents the steady state of a system.

Applications

The maximum modulus principle has many uses in complex analysis, and may be used to prove the following:

The fundamental theorem of algebra.
Schwarz's lemma, a result which in turn has many generalisations and applications in complex analysis.
The Phragmén–Lindelöf principle, an extension to unbounded domains.
The Borel–Carathéodory theorem, which bounds an analytic function in terms of its real part.
The Hadamard three-lines theorem, a result about the behaviour of bounded holomorphic functions on a line between two other parallel lines in the complex plane.

References

(See chapter 5.)
Conway, John B. (1978). Axler, S.; Gehring, F.W.; Ribet, K.A. (eds.). Functions of One Complex Variable I (2 ed.). New York: Springer Science+Business Media, Inc. ISBN 978-1-4612-6314-2.