generalized mean
Let ${x}_{1}$, ${x}_{2},\mathrm{\dots},{x}_{n}$ 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 $i=1,\mathrm{\dots},n$, then the generalized mean^{} is defined as
$${f}^{1}\left(\sum _{i=1}^{n}{p}_{i}f({x}_{i})\right).$$ 
Special cases

1.
$f(x)=x$, ${p}_{i}=1/n$ for all $i$: arithmetic mean^{}

2.
$f(x)=x$: weighted mean

3.
$f(x)=\mathrm{log}(x)$, ${p}_{i}=1/n$ 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 $i$: rootmeansquare^{}

6.
$f(x)={x}^{d}$ and ${p}_{i}=1/n$ for all $i$: power mean^{}

7.
$f(x)={x}^{d}$: weighted power mean

8.
$f(x)={2}^{(1\alpha )x}$, $\alpha \ne 1$, ${x}_{i}={\mathrm{log}}_{2}{p}_{i}$: Rényi’s $\alpha $entropy^{}
Title  generalized mean 

Canonical name  GeneralizedMean 
Date of creation  20130322 14:32:12 
Last modified on  20130322 14:32:12 
Owner  Mathprof (13753) 
Last modified by  Mathprof (13753) 
Numerical id  8 
Author  Mathprof (13753) 
Entry type  Definition 
Classification  msc 2600 
Synonym  KolmogorovNagumo function^{} of the mean 
Synonym  Hölder mean 