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# generalized mean

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

$f^{{-1}}\Big(\sum_{{i=1}}^{n}p_{i}f(x_{i})\Big).$ |

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. 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

Synonym:

Kolmogorov-Nagumo function of the mean,H\"older mean

Type of Math Object:

Definition

Major Section:

Reference

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26-00*no label found*

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## Comments

## generalized means inequality?

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.