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In mathematics, **generalized means** (or **power mean** or **Hölder mean** from Otto Hölder)^{[1]} are a family of functions for aggregating sets of numbers. These include as special cases the Pythagorean means (arithmetic, geometric, and harmonic means).

## Definition

If *p* is a non-zero real number, and are positive real numbers, then the **generalized mean** or **power mean** with exponent *p* of these positive real numbers is:^{[2]}

(See *p*-norm). For *p* = 0 we set it equal to the geometric mean (which is the limit of means with exponents approaching zero, as proved below):

Furthermore, for a sequence of positive weights *w*_{i} with sum we define the **weighted power mean** as:

The unweighted means correspond to setting all *w _{i}* = 1/

*n*.

## Special cases

A few particular values of yield special cases with their own names:^{[3]}

minimum | |

harmonic mean | |

geometric mean | |

arithmetic mean | |

root mean square or quadratic mean ^{[4]}^{[5]} | |

cubic mean | |

maximum |

Proof of (geometric mean) We can rewrite the definition of

*M*using the exponential function_{p}In the limit

*p*→ 0, we can apply L'Hôpital's rule to the argument of the exponential function. Differentiating the numerator and denominator with respect to*p*, we haveBy the continuity of the exponential function, we can substitute back into the above relation to obtain

as desired.

^{[2]}

Proof of and Assume (possibly after relabeling and combining terms together) that . Then

The formula for follows from

## Properties

Let be a sequence of positive real numbers, then the following properties hold:^{[1]}

- .
- Each generalized mean always lies between the smallest and largest of the
*x*values.

- Each generalized mean always lies between the smallest and largest of the
- , where is a permutation operator.
- Each generalized mean is a symmetric function of its arguments; permuting the arguments of a generalized mean does not change its value.

- .
- Like most means, the generalized mean is a homogeneous function of its arguments
*x*_{1}, ...,*x*. That is, if_{n}*b*is a positive real number, then the generalized mean with exponent*p*of the numbers is equal to*b*times the generalized mean of the numbers*x*_{1}, …,*x*._{n}

- Like most means, the generalized mean is a homogeneous function of its arguments
- .
- Like the quasi-arithmetic means, the computation of the mean can be split into computations of equal sized sub-blocks. This enables use of a divide and conquer algorithm to calculate the means, when desirable.

### Generalized mean inequality

In general,

- if
*p*<*q*, then

and the two means are equal if and only if *x*_{1} = *x*_{2} = ... = *x _{n}*.

The inequality is true for real values of *p* and *q*, as well as positive and negative infinity values.

It follows from the fact that, for all real *p*,

which can be proved using Jensen's inequality.

In particular, for *p* in {−1, 0, 1}, the generalized mean inequality implies the Pythagorean means inequality as well as the inequality of arithmetic and geometric means.

## Proof of power means inequality

We will prove weighted power means inequality, for the purpose of the proof we will assume the following without loss of generality:

Proof for unweighted power means is easily obtained by substituting *w _{i}* = 1/

*n*.

### Equivalence of inequalities between means of opposite signs

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

applying this, 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 mean

For any *q* > 0 and non-negative weights summing to 1, the following inequality holds:

The proof follows from Jensen's inequality, making use of the fact the logarithm is concave:

By applying the exponential function to both sides and observing that as a strictly increasing function it preserves the sign of the inequality, we get

Taking *q*th powers of the *x*_{i}, we are done for the inequality with positive *q*; the case for negatives is identical.

### Inequality between any two power means

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* : **R**_{+} → **R**_{+} . *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 replacing them with −*q* and −*p*, respectively.

## Generalized *f*-mean

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

This covers the geometric mean without using a limit with *f*(*x*) = *log*(*x*). The power mean is obtained for *f*(*x*) = *x ^{p}*.

## Applications

### Signal processing

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`

one 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 as an envelope detector on a rectified signal. - For small
*p*it can serve as a baseline detector on a mass spectrum.

## See also

- Arithmetic–geometric mean
- Average
- Heronian mean
- Inequality of arithmetic and geometric means
- Lehmer mean – also a mean related to powers
- Root mean square
- Minkowski distance

## Notes

- ^
^{a}^{b}Sýkora, Stanislav (2009).*Mathematical means and averages: basic properties*.**3**. Stan’s Library: Castano Primo, Italy. doi:10.3247/SL3Math09.001. - ^
^{a}^{b}P. S. Bullen:*Handbook of Means and Their Inequalities*. Dordrecht, Netherlands: Kluwer, 2003, pp. 175-177 **^**Weisstein, Eric W. "Power Mean".*MathWorld*. (retrieved 2019-08-17)**^**Thompson, Sylvanus P. (1965).*Calculus Made Easy*. Macmillan International Higher Education. p. 185. ISBN 9781349004874. Retrieved 5 July 2020.**^**Jones, Alan R. (2018).*Probability, Statistics and Other Frightening Stuff*. Routledge. p. 48. ISBN 9781351661386. Retrieved 5 July 2020.**^**If AC =*a*and BC =*b*. OC =**AM**of*a*and*b*, and radius*r*= QO = OG.

Using Pythagoras' theorem, QC² = QO² + OC² ∴ QC = √QO² + OC² =**QM**.

Using Pythagoras' theorem, OC² = OG² + GC² ∴ GC = √OC² − OG² =**GM**.

Using similar triangles, HC/GC = GC/OC ∴ HC = GC²/OC =**HM**.

## References and further reading

- P. S. Bullen:
*Handbook of Means and Their Inequalities*. Dordrecht, Netherlands: Kluwer, 2003, chapter III (The Power Means), pp. 175-265