generalized mean

Let x1, x2,,xn be real numbers, and f a continuousMathworldPlanetmath and strictly increasing or decreasing function on the real numbers. If each number xi is assigned a weight pi, with i=1npi=1, for i=1,,n, then the generalized meanMathworldPlanetmath is defined as


Special cases

  1. 1.

    f(x)=x, pi=1/n for all i: arithmetic meanMathworldPlanetmath

  2. 2.

    f(x)=x: weighted mean

  3. 3.

    f(x)=log(x), pi=1/n for all i: geometric meanMathworldPlanetmath

  4. 4.

    f(x)=1/x and pi=1/n for all i: harmonic meanMathworldPlanetmath

  5. 5.

    f(x)=x2 and pi=1/n for all i: root-mean-squareMathworldPlanetmathPlanetmath

  6. 6.

    f(x)=xd and pi=1/n for all i: power meanMathworldPlanetmath

  7. 7.

    f(x)=xd: weighted power mean

  8. 8.

    f(x)=2(1-α)x, α1, xi=-log2pi: Rényi’s α-entropyMathworldPlanetmath

Title generalized mean
Canonical name GeneralizedMean
Date of creation 2013-03-22 14:32:12
Last modified on 2013-03-22 14:32:12
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 8
Author Mathprof (13753)
Entry type Definition
Classification msc 26-00
Synonym Kolmogorov-Nagumo functionMathworldPlanetmath of the mean
Synonym Hölder mean