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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.

## DefinitionEdit

If is a non-zero real number, we can define the **generalized mean with exponent ** of the positive real numbers as

## PropertiesEdit

- Like most means, the generalized mean is a homogeneous function of its arguments . That is, if is a positive real number, then the generalized mean with exponent of the numbers is equal to times the generalized mean of the numbers .
- Like the quasi-arithmetic means, the computation of the mean can be split into computations of equal sized sub-blocks.

### Generalized mean inequality Edit

In general, if , then and the two means are equal if and only if . This follows from the fact that , which can be proved using Jensen's inequality.

In particular, for , the generalized mean inequality implies the Pythagorean means inequality as well as the inequality of arithmetic and geometric means.

## Special cases Edit

- - minimum,
- - harmonic mean,
- - geometric mean,
- - arithmetic mean,
- - quadratic mean,
- - maximum.

## Proof of power means inequalityEdit

### Equivalence of inequalities between means of opposite signsEdit

Suppose an average between power means with exponents *p* and *q* holds:

then:

We raise both sides to the power of -1 (strictly decreasing function in positive reals):

We get the inequality for means with exponents -*p* and -*q*, and we can use the same reasoning backwards, thus proving the inequalities to be equivalent, which will be used in some of the later proofs.

### Geometric meanEdit

For any *q* the inequality between mean with exponent *q* and geometric mean can be transformed in the following way:

(the first inequality is to be proven for positive *q*, and the latter otherwise)

We raise both sides to the power of *q*:

in both cases we get the inequality between weighted arithmetic and geometric means for the sequence , which can be proved by Jensen's inequality, making use of the fact the logarithmic function is concave:

By applying (strictly increasing) exp function to both sides we get the inequality:

Thus for any positive *q* it is true that:

since the inequality holds for any *q*, however small, and, as will be shown later, the expressions on the left and right approximate the geometric mean better as *q* approaches 0, the limit of the power mean for *q* approaching 0 is the geometric mean:

### Inequality between any two power meansEdit

We are to prove that for any *p*<*q* the following inequality holds:

if *p* is negative, and *q* is positive, the inequality is equivalent to the one proved above:

The proof for positive *p* and *q* is as follows:
Define the following function: . *f* is a power function, so it does have a second derivative: which is strictly positive within the domain of *f*, since *q* > *p*, so we know *f* is convex.

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

after raising both side to the power of 1/*q* (an increasing function, since 1/q is positive) we get the inequality which was to be proven:

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

### Minimum and maximumEdit

Minimum and maximum are assumed to be the power means with exponents of . Thus for any *q*:

For maximum the proof is as follows:
Assume WLoG that the sequence *x _{i}* is nonincreasing and no weight is zero.

Then the inequality is equivalent to:

After raising both sides to the power of *q* we get (depending on the sign of *q*) one of the inequalities:

≤ for *q*>0, ≥ for *q*<0.

After subtracting from the both sides we get:

After dividing by :

1 - *w*_{1} is nonzero, thus:

Substacting *x*_{1}^{q} leaves:

which is obvious, since *x*_{1} is greater or equal to any *x _{i}*, and thus:

For minimum the proof is almost the same, only instead of *x*_{1}, *w*_{1} we use *x*_{n}, *w*_{n}, QED.

## Generalized -mean Edit

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

which covers e.g. the geometric mean without using a limit. The power mean is obtained for .

## Applications Edit

### Signal processing Edit

A power mean serves a non-linear moving average
which is shifted towards small signal values for small
and emphasizes big signal values for big .
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 it can serve an envelope detector on a rectified signal.
- For small it can serve an baseline detector on a mass spectrum.

## See alsoEdit

- 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

## External linksEdit

- Power mean at MathWorld
- Examples of Generalized Mean
- A proof of the Generalized Mean on PlanetMath
- Rational Mean

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