derivation of zeroth weighted power mean

Let x1,x2,,xn be positive real numbers, and let w1,w2,,wn be positive real numbers such that w1+w2++wn=1. For r0, the r-th weighted power mean of x1,x2,,xn is


Using the Taylor seriesMathworldPlanetmath expansion et=1+t+𝒪(t2), where 𝒪(t2) is Landau notationMathworldPlanetmathPlanetmath for terms of order t2 and higher, we can write xir as


By substituting this into the definition of Mwr, we get

Mwr(x1,x2,,xn) = [w1(1+rlogx1)++wn(1+rlogxn)+𝒪(r2)]1/r
= [1+r(w1logx1++wnlogxn)+𝒪(r2)]1/r
= [1+rlog(x1w1x2w2xnwn)+𝒪(r2)]1/r
= exp{1rlog[1+rlog(x1w1x2w2xnwn)+𝒪(r2)]}.

Again using a Taylor series, this time log(1+t)=t+𝒪(t2), we get

Mwr(x1,x2,,xn) = exp{1r[rlog(x1w1x2w2xnwn)+𝒪(r2)]}
= exp[log(x1w1x2w2xnwn)+𝒪(r)].

Taking the limit r0, we find

Mw0(x1,x2,,xn) = exp[log(x1w1x2w2xnwn)]
= x1w1x2w2xnwn.

In particular, if we choose all the weights to be 1n,


the geometric meanMathworldPlanetmath of x1,x2,,xn.

Title derivation of zeroth weighted power mean
Canonical name DerivationOfZerothWeightedPowerMean
Date of creation 2013-03-22 13:10:29
Last modified on 2013-03-22 13:10:29
Owner pbruin (1001)
Last modified by pbruin (1001)
Numerical id 6
Author pbruin (1001)
Entry type Derivation
Classification msc 26B99
Related topic PowerMean
Related topic GeometricMean
Related topic GeneralMeansInequality
Related topic DerivationOfHarmonicMeanAsTheLimitOfThePowerMean