power mean

The $r$-th power mean of the numbers $x_{1},x_{2},\ldots,x_{n}$ is defined as:

 $M^{r}(x_{1},x_{2},\ldots,x_{n})=\left(\frac{x_{1}^{r}+x_{2}^{r}+\cdots+x_{n}^{% r}}{n}\right)^{1/r}.$

The arithmetic mean is a special case when $r=1$. The power mean is a continuous function of $r$, and taking limit when $r\to 0$ gives us the geometric mean:

 $M^{0}(x_{1},x_{2},\ldots,x_{n})=\sqrt[n]{x_{1}{x_{2}}\cdots x_{n}}.$

Also, when $r=-1$ we get

 $M^{-1}(x_{1},x_{2},\ldots,x_{n})=\frac{n}{\frac{1}{x_{1}}+\frac{1}{x_{2}}+% \cdots+\frac{1}{x_{n}}}$

the harmonic mean.

A generalization of power means are weighted power means.

 Title power mean Canonical name PowerMean Date of creation 2013-03-22 11:47:17 Last modified on 2013-03-22 11:47:17 Owner drini (3) Last modified by drini (3) Numerical id 14 Author drini (3) Entry type Definition Classification msc 26D15 Classification msc 16D10 Classification msc 00-01 Classification msc 34-00 Classification msc 35-00 Related topic WeightedPowerMean Related topic ArithmeticGeometricMeansInequality Related topic ArithmeticMean Related topic GeometricMean Related topic HarmonicMean Related topic GeneralMeansInequality Related topic RootMeanSquare3 Related topic ProofOfGeneralMeansInequality Related topic DerivationOfZerothWeightedPowerMean Related topic DerivationOfHarmonicMeanAsTheLimitOfThePowerMean