# general means inequality

If $0\neq r\in\mathbbmss{R}$, the $r$-mean (or $r$-th power mean) of the nonnegative numbers $a_{1},\ldots,a_{n}$ is defined as

 $M^{r}(a_{1},a_{2},\ldots,a_{n})=\left(\frac{1}{n}\displaystyle{\sum_{k=1}^{n}a% _{k}^{r}}\right)^{1/r}$

Given real numbers $x,y$ such that $xy\neq 0$ and $x, we have

 $M^{x}\leq M^{y}$

and the equality holds if and only if $a_{1}=...=a_{n}$.

Additionally, if we define $M^{0}$ to be the geometric mean $(a_{1}a_{2}...a_{n})^{1/n}$, we have that the inequality above holds for arbitrary real numbers $x.

The mentioned inequality is a special case of this one, since $M^{1}$ is the arithmetic mean, $M^{0}$ is the geometric mean and $M^{-1}$ is the harmonic mean.

This inequality can be further generalized using weighted power means.

 Title general means inequality Canonical name GeneralMeansInequality Date of creation 2013-03-22 12:39:49 Last modified on 2013-03-22 12:39:49 Owner drini (3) Last modified by drini (3) Numerical id 6 Author drini (3) Entry type Theorem Classification msc 26D15 Synonym power means inequality Related topic ArithmeticGeometricMeansInequality Related topic ArithmeticMean Related topic GeometricMean Related topic HarmonicMean Related topic PowerMean Related topic ProofOfArithmeticGeometricHarmonicMeansI Related topic RootMeanSquare3 Related topic DerivationOfZerothWeightedPowerMean Related topic ProofOfArithmeticGeometricHarmonicMeansInequality Related topic ComparisonOfPythagor