weighted power mean

If $w_1,w_2,\mathrm{\dots },w_n$ are positive real numbers such that $w_1+w_2+\mathrm{\cdots }+w_n=1$, we define the $r$-th weighted power mean of the $x_i$ as:

$$M_w^r(x_1,x_2,\mathrm{\dots },x_n)={\left(w_1{x}_1^r+w_2{x}_2^r+\mathrm{\cdots }+w_n{x}_n^r\right)}^{1/r}.$$

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

$$M_w^0=x_1^{w_1}{x}_2^{w_2}\mathrm{\cdots }{w}_n^{w_n}.$$

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

$$M_w^r\le M_w^s.$$

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