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Maximum modulus principle

Theorem that a holomorphic function’s absolute value cannot reach a strict local maximum at the interior of its domain

In mathematics, the maximum modulus principle in complex analysis states that if f {\displaystyle f} is a holomorphic function, then the modulus | f | {\displaystyle |f|} cannot exhibit a strict maximum that is strictly within the domain of f {\displaystyle f} . In other words, either f {\displaystyle f} is locally a constant function, or, for any point z 0 {\displaystyle z_{0}} inside the domain of f {\displaystyle f} there exist other points arbitrarily close to z 0 {\displaystyle z_{0}} at which | f | {\displaystyle |f|} takes larger values.

From Wikipedia

In mathematics, the maximum modulus principle in complex analysis states that if f {\displaystyle f} is a holomorphic function, then the modulus | f | {\displaystyle |f|} cannot exhibit a strict maximum that is strictly within the domain of f {\displaystyle f} . In other words, either f {\displaystyle f} is locally a constant function, or, for any point z 0 {\displaystyle z_{0}} inside the domain of f {\displaystyle f} there exist other points arbitrarily close to z 0 {\displaystyle z_{0}} at which | f | {\displaystyle |f|} takes larger values.

Text: Wikipédia, CC BY-SA 4.0. · Image: Oleg Alexandrov (Public domain) ·

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