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 $r<s$ 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