arithmetic-geometric-harmonic means inequality

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

max{x1,x2,,xn} x1+x2++xnn

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.

Title arithmetic-geometric-harmonic means inequality
Canonical name ArithmeticgeometricharmonicMeansInequality
Date of creation 2013-03-22 11:42:32
Last modified on 2013-03-22 11:42:32
Owner drini (3)
Last modified by drini (3)
Numerical id 22
Author drini (3)
Entry type TheoremMathworldPlanetmath
Classification msc 00A05
Classification msc 20-XX
Classification msc 26D15
Synonym harmonic-geometric-arithmetic means inequality
Synonym arithmetic-geometric means inequality
Synonym AGM inequality
Synonym AGMH inequality
Related topic ArithmeticMean
Related topic GeometricMean
Related topic HarmonicMean
Related topic GeneralMeansInequality
Related topic WeightedPowerMean
Related topic PowerMean
Related topic RootMeanSquare3
Related topic ProofOfGeneralMeansInequality
Related topic JensensInequality
Related topic DerivationOfHarmonicMeanAsTheLimitOfThePowerMean
Related topic MinimalAndMaximalNumber
Related topic ProofOfArithm