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generalized mean (Definition)

Definition

Let $ x_1$, $ x_2,\ldots, x_n$ be real numbers, and $ f$ a continuous and strictly increasing or decreasing function on the real numbers. If each number $ x_i$ is assigned a weight $ p_i$, with $ \sum_{i=1}^n p_i= 1$, for $ i=1,\ldots,n$, then the generalized mean is defined as

$\displaystyle f^{-1}\Big( \sum_{i=1}^n p_i f(x_i) \Big). $

Special cases

  1. $ f(x)=x$, $ p_i=1/n$ for all $ i$: arithmetic mean
  2. $ f(x)=x$: weighted mean
  3. $ f(x)=\log(x)$, $ p_i=1/n$ for all $ i$: geometric mean
  4. $ f(x)=1/x$ and $ p_i=1/n$ for all $ i$: harmonic mean
  5. $ f(x)=x^2$ and $ p_i=1/n$ for all $ i$: root-mean-square
  6. $ f(x)=x^d$ and $ p_i=1/n$ for all $ i$: power mean
  7. $ f(x)=x^d$: weighted power mean
  8. $ f(x)=2^{(1-\alpha)x}$, $ \alpha\neq 1$, $ x_i=-\log_2 p_i$: Rényi's $ \alpha$-entropy



"generalized mean" is owned by Mathprof. [ full author list (2) | owner history (1) ]
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Other names:  Kolmogorov-Nagumo function of the mean, Hölder mean
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Cross-references: weighted power mean, power mean, root-mean-square, harmonic mean, geometric mean, weighted mean, arithmetic mean, weight, number, function, decreasing, strictly increasing, continuous, real numbers
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This is version 5 of generalized mean, born on 2004-08-06, modified 2006-10-24.
Object id is 6081, canonical name is GeneralizedMean.
Accessed 3730 times total.

Classification:
AMS MSC: 26-00 (Real functions :: General reference works )
Pending Errata and Addenda
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generalized means inequality? by igor on 2004-08-09 11:13:30
Would it be possible to extend geometric-arithmetic-harmonic means
inequality to the case of generalized means?

I would foresee something like: given some conditions on f(x) and
g(x), the respective generalized means satisfy some given inequality.
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