# weighted power mean

If $w_{1},w_{2},\ldots,w_{n}$ are positive real numbers such that $w_{1}+w_{2}+\cdots+w_{n}=1$, we define the $r$-th weighted power mean of the $x_{i}$ as:

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

When all the $w_{i}=\frac{1}{n}$ we get the standard power mean. The weighted power mean is a continuous function of $r$, and taking limit when $r\to 0$ gives us

 $M_{w}^{0}=x_{1}^{w_{1}}x_{2}^{w_{2}}\cdots w_{n}^{w_{n}}.$

We can weighted use power means to generalize the power means inequality: If $w$ is a set of weights, and if $r then

 $M_{w}^{r}\leq M_{w}^{s}.$
 Title weighted power mean Canonical name WeightedPowerMean Date of creation 2013-03-22 11:47:20 Last modified on 2013-03-22 11:47:20 Owner drini (3) Last modified by drini (3) Numerical id 12 Author drini (3) Entry type Definition Classification msc 26B99 Classification msc 00-01 Classification msc 26-00 Related topic ArithmeticGeometricMeansInequality Related topic ArithmeticMean Related topic GeometricMean Related topic HarmonicMean Related topic PowerMean Related topic ProofOfArithmeticGeometricHarmonicMeansInequality Related topic RootMeanSquare3 Related topic ProofOfGeneralMeansInequality Related topic DerivationOfHarmonicMeanAsTheLimitOfThePowerMean