# 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. 1.

$f(x)=x$, $p_{i}=1/n$ for all $i$: arithmetic mean

2. 2.

$f(x)=x$: weighted mean

3. 3.

$f(x)=\log(x)$, $p_{i}=1/n$ for all $i$: geometric mean

4. 4.

$f(x)=1/x$ and $p_{i}=1/n$ for all $i$: harmonic mean

5. 5.

$f(x)=x^{2}$ and $p_{i}=1/n$ for all $i$: root-mean-square

6. 6.

$f(x)=x^{d}$ and $p_{i}=1/n$ for all $i$: power mean

7. 7.

$f(x)=x^{d}$: weighted power mean

8. 8.

$f(x)=2^{(1-\alpha)x}$, $\alpha\neq 1$, $x_{i}=-\log_{2}p_{i}$: Rényi’s $\alpha$-entropy

Title generalized mean GeneralizedMean 2013-03-22 14:32:12 2013-03-22 14:32:12 Mathprof (13753) Mathprof (13753) 8 Mathprof (13753) Definition msc 26-00 Kolmogorov-Nagumo function of the mean Hölder mean