Editing Maximum modulus principle (section)

==Formal statement==
Let <math>f</math> be a holomorphic function on some [[connected set|connected]] [[open set|open]] [[subset]] <math>D</math> of the [[complex plane]] <math>\mathbb{C}</math> and taking complex values. If <math>z_0</math> is a point in <math>D</math> such that 
:<math>|f(z_0)|\ge |f(z)|</math> 
for all <math>z</math> in some [[neighborhood (topology)|neighborhood]] of <math>z_0</math>, then <math>f</math> is constant on <math>D</math>.

This statement can be viewed as a special case of the [[open mapping theorem (complex analysis)|open mapping theorem]], which states that a nonconstant holomorphic function maps open sets to open sets: If <math>|f|</math> attains a local maximum at <math>z</math>, then the image of a sufficiently small open neighborhood of <math>z</math> cannot be open, so <math>f</math> is constant.

===Related statement===

Suppose that <math>D</math> is a bounded nonempty connected open subset of <math>\mathbb{C}</math>.
Let <math>\overline{D}</math> be the closure of <math>D</math>.
Suppose that <math>f \colon \overline{D} \to \mathbb{C}</math> is a continuous function that is holomorphic on <math>D</math>.
Then <math>|f(z)|</math> attains a maximum at some point of the boundary of <math>D</math>.

This follows from the first version as follows. Since <math>\overline{D}</math> is [[compact space|compact]] and nonempty, the continuous function <math>|f(z)|</math> attains a maximum at some point <math>z_0</math> of <math>\overline{D}</math>. If <math>z_0</math> is not on the boundary, then the maximum modulus principle implies that <math>f</math> is constant, so <math>|f(z)|</math> also attains the same maximum at any point of the boundary.

===Minimum modulus principle===

For a holomorphic function <math>f</math> on a connected open set <math>D</math> of <math>\mathbb{C}</math>, if <math>z_0</math> is a point in <math>D</math> such that
:<math>0 < |f(z_0)| \le |f(z)|</math>
for all <math>z</math> in some [[neighborhood (topology)|neighborhood]] of <math>z_0</math>, then <math>f</math> is constant on <math>D</math>.

Proof: Apply the maximum modulus principle to <math>1/f</math>.