generalized mean

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

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

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 function of the mean
Synonym	Hölder mean