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A generalized mean, also known as power mean or Hölder mean, is an abstraction of the Pythagorean means including arithmetic, geometric, and harmonic means.
## Definition

## Properties

### Generalized mean inequality

## Special cases

## Proof of power means inequality

### Equivalence of inequalities between means of opposite signs

Suppose an average between power means with exponents p and q holds:
### Geometric mean

For any q the inequality between mean with exponent q and geometric mean can be transformed in the following way:
### Inequality between any two power means

We are to prove that for any p<q the following inequality holds:
### Minimum and maximum

Minimum and maximum are assumed to be the power means with exponents of
$-infty$ and $+infty$. Thus for any q:
_{i} is nonincreasing and no weight is zero._{1} is nonzero, thus:
_{1}^{q} leaves:
_{1} is greater or equal to any x_{i}, and thus:
## Generalized $f$-mean

## Applications

### Signal processing

## See also

## External links

If $p$ is a non-zero real number, we can define the generalized mean with exponent $p$ of the positive real numbers $x\_1,dots,x\_n$ as

- $$

- Like most means, the generalized mean is a homogeneous function of its arguments $x\_1,dots,x\_n$. That is, if $b$ is a positive real number, then the generalized mean with exponent $p$ of the numbers $bcdot\; x\_1,dots,\; bcdot\; x\_n$ is equal to $b$ times the generalized mean of the numbers $x\_1,dots,\; x\_n$.
- Like the quasi-arithmetic means, the computation of the mean can be split into computations of equal sized sub-blocks.

- $$

dots,M_p(x_{(n-1)cdot k + 1},dots,x_{ncdot k}))

In general, if $p\; <\; q$, then $M\_p(x\_1,dots,x\_n)\; le\; M\_q(x\_1,dots,x\_n)$ and the two means are equal if and only if $x\_1\; =\; x\_2\; =\; cdots\; =\; x\_n$. This follows from the fact that

- $forall\; pinmathbb\{R\}\; frac\{partial\; M\_p(x\_1,dots,x\_n)\}\{partial\; p\}geq\; 0,$

which can be proved using Jensen's inequality.

In particular, for $pin\{-1,\; 0,\; 1\}$, the generalized mean inequality implies the Pythagorean means inequality as well as the inequality of arithmetic and geometric means.

- $lim\_\{pto-infty\}\; M\_p(x\_1,dots,x\_n)\; =\; min\; \{x\_1,dots,x\_n\}$ - minimum,
- $M\_\{-1\}(x\_1,dots,x\_n)\; =\; frac\{n\}\{frac\{1\}\{x\_1\}+dots+frac\{1\}\{x\_n\}\}$ - harmonic mean,
- $lim\_\{pto0\}\; M\_p(x\_1,dots,x\_n)\; =\; sqrt[n]\{x\_1cdotdotscdot\; x\_n\}$ - geometric mean,
- $M\_1(x\_1,dots,x\_n)\; =\; frac\{x\_1\; +\; dots\; +\; x\_n\}\{n\}$ - arithmetic mean,
- $M\_2(x\_1,dots,x\_n)\; =\; sqrt\{frac\{x\_1^2\; +\; dots\; +\; x\_n^2\}\{n\}\}$ - quadratic mean,
- $lim\_\{ptoinfty\}\; M\_p(x\_1,dots,x\_n)\; =\; max\; \{x\_1,dots,x\_n\}$ - maximum.

- $sqrt[p]\{sum\_\{i=1\}^nw\_ix\_i^p\}leq\; sqrt[q]\{sum\_\{i=1\}^nw\_ix\_i^q\}$

- $sqrt[p]\{sum\_\{i=1\}^nfrac\{w\_i\}\{x\_i^p\}\}leq\; sqrt[q]\{sum\_\{i=1\}^nfrac\{w\_i\}\{x\_i^q\}\}$

- $sqrt[-p]\{sum\_\{i=1\}^nw\_ix\_i^\{-p\}\}=sqrt[p]\{frac\{1\}\{sum\_\{i=1\}^nw\_ifrac\{1\}\{x\_i^p\}\}\}geq\; sqrt[q]\{frac\{1\}\{sum\_\{i=1\}^nw\_ifrac\{1\}\{x\_i^q\}\}\}=sqrt[-q]\{sum\_\{i=1\}^nw\_ix\_i^\{-q\}\}$

- $prod\_\{i=1\}^nx\_i^\{w\_i\}\; leq\; sqrt[q]\{sum\_\{i=1\}^nw\_ix\_i^q\}$

- $sqrt[q]\{sum\_\{i=1\}^nw\_ix\_i^q\}leq\; prod\_\{i=1\}^nx\_i^\{w\_i\}$

We raise both sides to the power of q:

- $prod\_\{i=1\}^nx\_i^\{w\_icdot\; q\}\; leq\; sum\_\{i=1\}^nw\_ix\_i^q$

- $sum\_\{i=1\}^nw\_ilog(x\_i)\; leq\; log(sum\_\{i=1\}^nw\_ix\_i)$

- $log(prod\_\{i=1\}^nx\_i^\{w\_i\})\; leq\; log(sum\_\{i=1\}^nw\_ix\_i)$

- $prod\_\{i=1\}^nx\_i^\{w\_i\}\; leq\; sum\_\{i=1\}^nw\_ix\_i$

Thus for any positive q it is true that:

- $sqrt[-q]\{sum\_\{i=1\}^nw\_ix\_i^\{-q\}\}leq\; prod\_\{i=1\}^nx\_i^\{w\_i\}\; leq\; sqrt[q]\{sum\_\{i=1\}^nw\_ix\_i^q\}$

- $lim\_\{qrightarrow\; 0\}sqrt[q]\{sum\_\{i=1\}^nw\_ix\_i^\{q\}\}=prod\_\{i=1\}^nx\_i^\{w\_i\}$

- $sqrt[p]\{sum\_\{i=1\}^nw\_ix\_i^p\}leq\; sqrt[q]\{sum\_\{i=1\}^nw\_ix\_i^q\}$

- $sqrt[p]\{sum\_\{i=1\}^nw\_ix\_i^p\}leq\; prod\_\{i=1\}^nx\_i^\{w\_i\}\; leqsqrt[q]\{sum\_\{i=1\}^nw\_ix\_i^q\}$

Using this, and the Jensen's inequality we get:

- $f(sum\_\{i=1\}^nw\_ix\_i^p)leqsum\_\{i=1\}^nw\_if(x\_i^p)$

- $sqrt[frac\{p\}\{q\}]\{sum\_\{i=1\}^nw\_ix\_i^p\}leqsum\_\{i=1\}^nw\_ix\_i^q$

- $sqrt[p]\{sum\_\{i=1\}^nw\_ix\_i^p\}leqsqrt[q]\{sum\_\{i=1\}^nw\_ix\_i^q\}$

Using the previously shown equivalence we can prove the inequality for negative p and q by substituting them with, respectively, -q and -p, QED.

- $min\; (x\_1,x\_2,ldots\; ,x\_n)leq\; sqrt[q]\{sum\_\{i=1\}^nw\_ix\_i^q\}leq\; max\; (x\_1,x\_2,ldots\; ,x\_n)$

Then the inequality is equivalent to:

- $sqrt[q]\{sum\_\{i=1\}^nw\_ix\_i^q\}leq\; x\_1$

- $sum\_\{i=1\}^nw\_ix\_i^qleq\; \{color\{red\}\; geq\}\; x\_1^q$

After subtracting $w\_1x\_1$ from the both sides we get:

- $sum\_\{i=2\}^nw\_ix\_i^qleq\; \{color\{red\}\; geq\}\; (1-w\_1)x\_1^q$

- $sum\_\{i=2\}^nfrac\{w\_i\}\{(1-w\_1)\}x\_i^qleq\; \{color\{red\}\; geq\}\; x\_1^q$

- $sum\_\{i=2\}^nfrac\{w\_i\}\{(1-w\_1)\}=1$

- $sum\_\{i=2\}^nfrac\{w\_i\}\{(1-w\_1)\}(x\_i^q-x\_1^q)leq\; \{color\{red\}\; geq\}\; 0$

- $x\_i^q-x\_1^qleq\; \{color\{red\}\; geq\}\; 0$

For minimum the proof is almost the same, only instead of x_{1}, w_{1} we use x_{n}, w_{n}, QED.

The power mean could be generalized further to the generalized $f$-mean:

- $M\_f(x\_1,dots,x\_n)\; =\; f^\{-1\}$

which covers e.g. the geometric mean without using a limit. The power mean is obtained for $fleft(xright)=x^p$.

A power mean serves a non-linear moving average
which is shifted towards small signal values for small $p$
and emphasizes big signal values for big $p$.
Given an efficient implementation of a moving arithmetic mean
called `smooth` you can implement a moving power mean
according to the following Haskell code.

powerSmooth :: Floating a => ([a] -> [a]) -> a -> [a] -> [a]

powerSmooth smooth p =

map (** recip p) . smooth . map (**p)

- For big $p$ it can serve an envelope detector on a rectified signal.
- For small $p$ it can serve an baseline detector on a mass spectrum.

- Inequality of arithmetic and geometric means
- arithmetic mean
- geometric mean
- harmonic mean
- Heronian mean
- Lehmer mean - also a mean related to powers
- average
- root mean square

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Last updated on Wednesday September 10, 2008 at 16:12:50 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Wednesday September 10, 2008 at 16:12:50 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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