Maximum modulus principle

In mathematics, the maximum modulus principle in complex analysis states that if

is locally a constant function, or, for any point

there exist other points arbitrarily close to

takes larger values.

be a holomorphic function on some connected open subset

and taking complex values.

This statement can be viewed as a special case of the open mapping theorem, which states that a nonconstant holomorphic function maps open sets to open sets: If

, then the image of a sufficiently small open neighborhood of

is a bounded nonempty connected open subset of

attains a maximum at some point of the boundary of

is compact and nonempty, the continuous function

is not on the boundary, then the maximum modulus principle implies that

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

Proof: Apply the maximum modulus principle to

One can use the equality for complex natural logarithms to deduce that

Similar reasoning shows that

can only have a local minimum (which necessarily has value 0) at an isolated zero of

Another proof works by using Gauss's mean value theorem to "force" all points within overlapping open disks to assume the same value as the maximum.

The disks are laid such that their centers form a polygonal path from the value where

(a closed ball centered at

We then define the boundary of the closed ball with positive orientation as

Invoking Cauchy's integral formula, we obtain For all

This also holds for all balls of radius less than

Then one can construct a sequence of distinct points located in

is closed, the sequence converges to some point in

A physical interpretation of this principle comes from the heat equation.

is harmonic, it is thus the steady state of a heat flow on the region

Suppose a strict maximum was attained on the interior of

, 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.

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

A plot of the modulus of (in red) for in the unit disk centered at the origin (shown in blue). As predicted by the theorem, the maximum of the modulus cannot be inside of the disk (so the highest value on the red surface is somewhere along its edge).