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单词 ArithmeticgeometricharmonicMeansInequality
释义

arithmetic-geometric-harmonic means inequality


Let x1,x2,,xn be positive numbers.Then

max{x1,x2,,xn}x1+x2++xnn
x1x2xnn
n1x1+1x2++1xn
min{x1,x2,,xn}

The equality is obtained if and only if x1=x2==xn.

There are several generalizationsPlanetmathPlanetmath to this inequalityMathworldPlanetmath using power meansMathworldPlanetmath and weighted power means.

Titlearithmetic-geometric-harmonic means inequality
Canonical nameArithmeticgeometricharmonicMeansInequality
Date of creation2013-03-22 11:42:32
Last modified on2013-03-22 11:42:32
Ownerdrini (3)
Last modified bydrini (3)
Numerical id22
Authordrini (3)
Entry typeTheoremMathworldPlanetmath
Classificationmsc 00A05
Classificationmsc 20-XX
Classificationmsc 26D15
Synonymharmonic-geometric-arithmetic means inequality
Synonymarithmetic-geometric means inequality
SynonymAGM inequality
SynonymAGMH inequality
Related topicArithmeticMean
Related topicGeometricMean
Related topicHarmonicMean
Related topicGeneralMeansInequality
Related topicWeightedPowerMean
Related topicPowerMean
Related topicRootMeanSquare3
Related topicProofOfGeneralMeansInequality
Related topicJensensInequality
Related topicDerivationOfHarmonicMeanAsTheLimitOfThePowerMean
Related topicMinimalAndMaximalNumber
Related topicProofOfArithm
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更新时间:2025/5/4 19:52:44