# arithmetic-geometric-harmonic means inequality

Let $x_{1},x_{2},\ldots,x_{n}$ be positive numbers. Then

 $\displaystyle\max\{x_{1},x_{2},\ldots,x_{n}\}$ $\displaystyle\geq$ $\displaystyle\frac{x_{1}+x_{2}+\cdots+x_{n}}{n}$ $\displaystyle\geq$ $\displaystyle\sqrt[n]{x_{1}x_{2}\cdots x_{n}}$ $\displaystyle\geq$ $\displaystyle\frac{n}{\frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots+\frac{1}{x_{n}}}$ $\displaystyle\geq$ $\displaystyle\min\{x_{1},x_{2},\ldots,x_{n}\}$

The equality is obtained if and only if $x_{1}=x_{2}=\cdots=x_{n}$.

There are several generalizations to this inequality using power means 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 Theorem 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