 TOPICS  # Maximum Modulus Principle

Let be a domain, and let be an analytic function on . Then if there is a point such that for all , then is constant. The following slightly sharper version can also be formulated. Let be a domain, and let be an analytic function on . Then if there is a point at which has a local maximum, then is constant.

Furthermore, let be a bounded domain, and let be a continuous function on the closed set that is analytic on . Then the maximum value of on (which always exists) occurs on the boundary . In other words, The maximum modulus theorem is not always true on an unbounded domain.

Complex Modulus, Minimum Modulus Principle

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## References

Krantz, S. G. "The Maximum Modulus Principle" and "Boundary Maximum Modulus Theorem." §5.4.1 and 5.4.2 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 76-77, 1999.

## Referenced on Wolfram|Alpha

Maximum Modulus Principle

## Cite this as:

Weisstein, Eric W. "Maximum Modulus Principle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MaximumModulusPrinciple.html