weighted power mean

If w1,w2,,wn are positive real numbers such that w1+w2++wn=1, we define the r-th weighted power mean of the xi as:


When all the wi=1n we get the standard power meanMathworldPlanetmath. The weighted power mean is a continuous functionMathworldPlanetmathPlanetmath of r, and taking limit when r0 gives us


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

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