Maximum-modulus theorem
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.
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
for all z in a neighborhood of z0, then the function f is constant on D.
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 |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 z, then the image of a sufficiently small open neighborhood of z cannot be open. Therefore, f is constant.
One can use the equality
for complex natural logarithms to deduce that ln |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.
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