Maximum modulus principle

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.

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 holomorphic function maps open sets to open sets. If |f| attains a local maximum at a, then clearly the direct image of sufficiently small open neighborhoods of a cannot be open. Therefore, f is constant.


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


  • E.C. Titchmarsh, The Theory of Functions (2nd Ed) (1939) Oxford University Press. (See chapter 5.)

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