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In mathematics, the maximum modulus principle in complex analysis states that if f is a holomorphic function, then the modulus |f| cannot exhibit a true local maximum that is properly within the domain of f.

In other words, either f is a constant function, or, for any point z0 inside the domain of f there exist other points arbitrarily close to z0 at which |f | takes larger values.

Formal statement

Let f be a function holomorphic on some connected open subset D of the complex plane C and taking complex values. If z0 is a point in D such that

$$|f(z_0)|\ge |f(z)|$$

for all z in a neighborhood of z0, then the function f is constant on D.

Sketch of the proof

One uses the equality

log f(z) = log |f(z)| + i arg f(z)

for complex natural logarithms to deduce that log |f(z)| is a harmonic function. Since z0 is a local maximum for this function also, it follows from the maximum principle that |f(z)| is constant. Then, using the Cauchy-Riemann equations we show that f'(z)=0, and thus that f(z) is constant as well.

Another proof works by using Gauss's mean value theorem to "force" all points within overlapping open disks to assume the same value. The disks are laid such that their centers form a polygonal path from the value where f(z) is maximized to any other point in the domain, while being totally contained within the domain. Thus the existence of a maximum value implies that all the values in the domain are the same, thus f(z) is constant.

By switching to the reciprocal, we can get the minimum modulus principle. It states that if f is holomorphic within a bounded domain D, continuous up to the boundary of D, and non-zero at all points, then the modulus |f (z)| takes its minimum value on the boundary of D.

Alternatively, the maximum modulus principle 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 |f| attains a local maximum at a, then the image of a sufficiently small open neighborhood of a cannot be open. Therefore, f is constant.
Physical Interpretation

A physical interpretation of this principle comes from the heat equation. That is, since log |f(z)| is harmonic, it is thus the steady state of a heat flow on the region D. Suppose a strict maximum was attained on the interior of D, 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.

References

E.C. Titchmarsh, The Theory of Functions (2nd Ed) (1939) Oxford University Press. (See chapter 5.)
E.D. Solomentsev (2001), "Maximum-modulus principle", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104