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.
One uses the equality
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.
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