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GeezerVison is showing mathworld.wolfram.com — Minimum Modulus Principle -- from Wolfram MathWorld

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Minimum Modulus Principle

The minimum modulus principle states that a nonvanishing analytic function attaining an interior minimum of its absolute value is constant. Let f be analytic on a domain U subset= C, and assume that f never vanishes. 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.

Let U subset= C be a bounded domain, let f be a continuous function on the closed set U^_ that is analytic on U, and assume that f never vanishes on U^_. Then the minimum value of |f| on U^_ (which always exists) must occur on partialU. In other words,

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

See also

Complex Modulus, Maximum Modulus Principle

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References

Krantz, S. G. "The Minimum Principle." §5.4.3 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 77, 1999.

Referenced on Wolfram|Alpha

Minimum Modulus Principle

Cite this as:

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

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