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Maximum Modulus Principle

Let U subset= C be a domain, and let f be an analytic function on U. Then if there is a point z_0 in U such that |f(z_0)|>=|f(z)| for all z in U, then f is constant. The following slightly sharper version can also be formulated. Let U subset= C be a domain, and let f be an analytic function on U. Then if there is a point z_0 in U at which |f| has a local maximum, then f is constant.

Furthermore, let U subset= C be a bounded domain, and let f be a continuous function on the closed set U^_ that is analytic on U. Then the maximum value of |f| on U^_ (which always exists) occurs on the boundary partialU. In other words,

max_(U^_)|f|=max_(partialU)|f|.

The maximum modulus theorem is not always true on an unbounded domain.

See also

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.

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Maximum Modulus Principle

Cite this as:

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

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