Title: Minimum Modulus Principle -- from Wolfram MathWorld
Open Graph Title: Minimum Modulus Principle -- from Wolfram MathWorld
X Title: Minimum Modulus Principle -- from Wolfram MathWorld
Description: 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|.
Open Graph Description: 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|.
X Description: 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|.
Opengraph URL: https://mathworld.wolfram.com/MinimumModulusPrinciple.html
Domain: mathworld.wolfram.com
| DC.Title | Minimum Modulus Principle |
| DC.Creator | Weisstein, Eric W. |
| DC.Description | 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|. |
| DC.Date.Created | 2000-06-20 |
| DC.Subject | 26 |
| DC.Rights | Copyright 1999-2026 Wolfram Research, Inc. See https://mathworld.wolfram.com/about/terms.html for a full terms of use statement. |
| DC.Format | text/html |
| DC.Identifier | https://mathworld.wolfram.com/MinimumModulusPrinciple.html |
| DC.Language | en |
| DC.Publisher | Wolfram Research, Inc. |
| DC.Relation.IsPartOf | https://mathworld.wolfram.com/ |
| DC.Type | Text |
| Last-Modified | 2000-06-20 |
| og:image | https://mathworld.wolfram.com/images/socialmedia/share.png |
| og:type | website |
| twitter:card | summary_large_image |
| twitter:image:src | https://mathworld.wolfram.com/images/socialmedia/share.png |
| None | ie=edge |
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