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单词 GeneralMeansInequality
释义

general means inequality


The power means inequality is a generalization of arithmetic-geometric means inequality.

If 0r, the r-mean (or r-th power meanMathworldPlanetmath) of the nonnegativenumbers a1,,an is defined as

Mr(a1,a2,,an)=(1nk=1nakr)1/r

Given real numbers x,y such that xy0and x<y, we have

MxMy

and the equality holds if and only if a1==an.

Additionally, if we define M0 to be thegeometric meanMathworldPlanetmath (a1a2an)1/n, we havethat the inequalityMathworldPlanetmath above holds for arbitrary real numbers x<y.

The mentioned inequality is a special case of this one, since M1 is the arithmetic meanMathworldPlanetmath, M0 is the geometric mean and M-1 is the harmonic meanMathworldPlanetmath.

This inequality can be further generalized using weighted power means.

Titlegeneral means inequality
Canonical nameGeneralMeansInequality
Date of creation2013-03-22 12:39:49
Last modified on2013-03-22 12:39:49
Ownerdrini (3)
Last modified bydrini (3)
Numerical id6
Authordrini (3)
Entry typeTheorem
Classificationmsc 26D15
Synonympower means inequality
Related topicArithmeticGeometricMeansInequality
Related topicArithmeticMean
Related topicGeometricMean
Related topicHarmonicMean
Related topicPowerMean
Related topicProofOfArithmeticGeometricHarmonicMeansI
Related topicRootMeanSquare3
Related topicDerivationOfZerothWeightedPowerMean
Related topicProofOfArithmeticGeometricHarmonicMeansInequality
Related topicComparisonOfPythagor
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更新时间:2025/5/4 16:34:48