power mean

The r-th power meanMathworldPlanetmath of the numbers x1,x2,,xn is defined as:


The arithmetic meanMathworldPlanetmath is a special case when r=1. The power mean is a continuous functionMathworldPlanetmathPlanetmath of r, and taking limit when r0 gives us the geometric meanMathworldPlanetmath:


Also, when r=-1 we get


the harmonic meanMathworldPlanetmath.

A generalizationPlanetmathPlanetmath 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