# Generalized mean

__: N-th root of the arithmetic mean of the given numbers raised to the power n__

**Short description**This article needs additional citations for verification. (June 2020) (Learn how and when to remove this template message) |

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 [math]\displaystyle{ x_1, \dots, x_n }[/math] are positive real numbers, then the **generalized mean** or **power mean** with exponent p of these positive real numbers is:^{[2]}

[math]\displaystyle{ M_p(x_1,\dots,x_n) = \left( \frac{1}{n} \sum_{i=1}^n x_i^p \right)^{{1}/{p}} . }[/math]

(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):

[math]\displaystyle{ M_0(x_1, \dots, x_n) = \left(\prod_{i=1}^n x_i\right)^{1/n} . }[/math]

Furthermore, for a sequence of positive weights w_{i} we define the **weighted power mean** as:^{[2]}
[math]\displaystyle{ M_p(x_1,\dots,x_n) = \left(\frac{\sum_{i=1}^n w_i x_i^p}{\sum_{i=1}^n w_i} \right)^{{1}/{p}} }[/math]
and when *p* = 0, it is equal to the weighted geometric mean:

[math]\displaystyle{ M_0(x_1,\dots,x_n) = \left(\prod_{i=1}^n x_i^{w_i}\right)^{1 / \sum_{i=1}^n w_i} . }[/math]

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

## Special cases

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

- minimum
- [math]\displaystyle{ M_{-\infty}(x_1,\dots,x_n) = \lim_{p\to-\infty} M_p(x_1,\dots,x_n) = \min \{x_1,\dots,x_n\} }[/math]
- harmonic mean
- [math]\displaystyle{ M_{-1}(x_1,\dots,x_n) = \frac{n}{\frac{1}{x_1}+\dots+\frac{1}{x_n}} }[/math]
- geometric mean
- [math]\displaystyle{ M_0(x_1,\dots,x_n) = \lim_{p\to0} M_p(x_1,\dots,x_n) = \sqrt[n]{x_1\cdot\dots\cdot x_n} }[/math]
- arithmetic mean
- [math]\displaystyle{ M_1(x_1,\dots,x_n) = \frac{x_1 + \dots + x_n}{n} }[/math]
- root mean square

or quadratic mean^{[4]}^{[5]} - [math]\displaystyle{ M_2(x_1,\dots,x_n) = \sqrt{\frac{x_1^2 + \dots + x_n^2}{n}} }[/math]
- cubic mean
- [math]\displaystyle{ M_3(x_1,\dots,x_n) = \sqrt[3]{\frac{x_1^3 + \dots + x_n^3}{n}} }[/math]
- maximum
- [math]\displaystyle{ M_{+\infty}(x_1,\dots,x_n) = \lim_{p\to\infty} M_p(x_1,\dots,x_n) = \max \{x_1,\dots,x_n\} }[/math]

Proof of [math]\displaystyle{ \lim_{p \to 0} M_p = M_0 }[/math] (geometric mean)
We can rewrite the definition of M_{p} using the exponential function

[math]\displaystyle{ M_p(x_1,\dots,x_n) = \exp{\left( \ln{\left[\left(\sum_{i=1}^n w_ix_{i}^p \right)^{1/p}\right]} \right) } = \exp{\left( \frac{\ln{\left(\sum_{i=1}^n w_ix_{i}^p \right)}}{p} \right) } }[/math]

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 have
[math]\displaystyle{ \begin{align}
\lim_{p \to 0} \frac{\ln{\left(\sum_{i=1}^n w_ix_{i}^p \right)}}{p} &= \lim_{p \to 0} \frac{\frac{\sum_{i=1}^n w_i x_i^p \ln{x_i}}{\sum_{j=1}^n w_j x_j^p}}{1} \\
&= \lim_{p \to 0} \frac{\sum_{i=1}^n w_i x_i^p \ln{x_i}}{\sum_{j=1}^n w_j x_j^p} \\
&= \sum_{i=1}^n \frac{ w_i \ln{x_i}}{ \lim_{p \to 0} \sum_{j=1}^n w_j \left( \frac{x_j}{x_i} \right)^p} \\
&= \sum_{i=1}^n w_i \ln{x_i} \\
&= \ln{\left(\prod_{i=1}^n x_i^{w_i} \right)}
\end{align} }[/math]

By the continuity of the exponential function, we can substitute back into the above relation to obtain
[math]\displaystyle{ \lim_{p \to 0} M_p(x_1,\dots,x_n) = \exp{\left( \ln{\left(\prod_{i=1}^n x_i^{w_i} \right)} \right)} = \prod_{i=1}^n x_i^{w_i} = M_0(x_1,\dots,x_n) }[/math]
as desired.^{[2]}

**Proof of [math]\displaystyle{ \lim_{p \to \infty} M_p = M_\infty }[/math] and [math]\displaystyle{ \lim_{p \to -\infty} M_p = M_{-\infty} }[/math]**

Assume (possibly after relabeling and combining terms together) that [math]\displaystyle{ x_1 \geq \dots \geq x_n }[/math]. Then

[math]\displaystyle{ \begin{align} \lim_{p \to \infty} M_p(x_1,\dots,x_n) &= \lim_{p \to \infty} \left( \sum_{i=1}^n w_i x_i^p \right)^{1/p} \\ &= x_1 \lim_{p \to \infty} \left( \sum_{i=1}^n w_i \left( \frac{x_i}{x_1} \right)^p \right)^{1/p} \\ &= x_1 = M_\infty (x_1,\dots,x_n). \end{align} }[/math]

The formula for [math]\displaystyle{ M_{-\infty} }[/math] follows from [math]\displaystyle{ M_{-\infty} (x_1,\dots,x_n) = \frac{1}{M_\infty (1/x_1,\dots,1/x_n)} = x_n. }[/math]

## Properties

Let [math]\displaystyle{ x_1, \dots, x_n }[/math] be a sequence of positive real numbers, then the following properties hold:^{[1]}

- [math]\displaystyle{ \min(x_1, \dots, x_n) \le M_p(x_1, \dots, x_n) \le \max(x_1, \dots, x_n) }[/math].
- [math]\displaystyle{ M_p(x_1, \dots, x_n) = M_p(P(x_1, \dots, x_n)) }[/math], where [math]\displaystyle{ P }[/math] is a permutation operator.
- [math]\displaystyle{ M_p(b x_1, \dots, b x_n) = b \cdot M_p(x_1, \dots, x_n) }[/math].
- [math]\displaystyle{ M_p(x_1, \dots, x_{n \cdot k}) = M_p\left[M_p(x_1, \dots, x_{k}), M_p(x_{k + 1}, \dots, x_{2 \cdot k}), \dots, M_p(x_{(n - 1) \cdot k + 1}, \dots, x_{n \cdot k})\right] }[/math].

### Generalized mean inequality

In general, if *p* < *q*, then
[math]\displaystyle{ M_p(x_1, \dots, x_n) \le M_q(x_1, \dots, x_n) }[/math]
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, [math]\displaystyle{ \frac{\partial}{\partial p}M_p(x_1, \dots, x_n) \geq 0 }[/math] 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: [math]\displaystyle{ \begin{align} w_i \in [0, 1] \\ \sum_{i=1}^nw_i = 1 \end{align} }[/math]

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: [math]\displaystyle{ \left(\sum_{i=1}^n w_i x_i^p\right)^{1/p} \geq \left(\sum_{i=1}^n w_i x_i^q\right)^{1/q} }[/math] applying this, then: [math]\displaystyle{ \left(\sum_{i=1}^n\frac{w_i}{x_i^p}\right)^{1/p} \geq \left(\sum_{i=1}^n\frac{w_i}{x_i^q}\right)^{1/q} }[/math]

We raise both sides to the power of −1 (strictly decreasing function in positive reals): [math]\displaystyle{ \left(\sum_{i=1}^nw_ix_i^{-p}\right)^{-1/p} = \left(\frac{1}{\sum_{i=1}^nw_i\frac{1}{x_i^p}}\right)^{1/p} \leq \left(\frac{1}{\sum_{i=1}^nw_i\frac{1}{x_i^q}}\right)^{1/q} = \left(\sum_{i=1}^nw_ix_i^{-q}\right)^{-1/q} }[/math]

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:
[math]\displaystyle{ \left(\sum_{i=1}^n w_i x_i^{-q}\right)^{-1/q} \leq \prod_{i=1}^n x_i^{w_i} \leq \left(\sum_{i=1}^n w_i x_i^q\right)^{1/q}. }[/math]

The proof follows from Jensen's inequality, making use of the fact the logarithm is concave: [math]\displaystyle{ \log \prod_{i=1}^n x_i^{w_i} = \sum_{i=1}^n w_i\log x_i \leq \log \sum_{i=1}^n w_i x_i. }[/math]

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 [math]\displaystyle{ \prod_{i=1}^n x_i^{w_i} \leq \sum_{i=1}^n w_i x_i. }[/math]

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:
[math]\displaystyle{ \left(\sum_{i=1}^n w_i x_i^p\right)^{1/p} \leq \left(\sum_{i=1}^nw_ix_i^q\right)^{1/q} }[/math]
if p is negative, and q is positive, the inequality is equivalent to the one proved above:
[math]\displaystyle{ \left(\sum_{i=1}^nw_i x_i^p\right)^{1/p} \leq \prod_{i=1}^n x_i^{w_i} \leq \left(\sum_{i=1}^n w_i x_i^q\right)^{1/q} }[/math]

The proof for positive p and q is as follows: Define the following function: *f* : **R**_{+} → **R**_{+} [math]\displaystyle{ f(x)=x^{\frac{q}{p}} }[/math]. f is a power function, so it does have a second derivative:
[math]\displaystyle{ f''(x) = \left(\frac{q}{p} \right) \left( \frac{q}{p}-1 \right)x^{\frac{q}{p}-2} }[/math]
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:
[math]\displaystyle{ \begin{align}
f \left( \sum_{i=1}^nw_ix_i^p \right) &\leq \sum_{i=1}^nw_if(x_i^p) \\[3pt]
\left(\sum_{i=1}^n w_i x_i^p\right)^{q/p} &\leq \sum_{i=1}^nw_ix_i^q
\end{align} }[/math]
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:

[math]\displaystyle{ \left(\sum_{i=1}^n w_i x_i^p\right)^{1/p} \leq \left(\sum_{i=1}^n w_i x_i^q\right)^{1/q} }[/math]

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:

[math]\displaystyle{ M_f(x_1,\dots,x_n) = f^{-1} \left({\frac{1}{n}\cdot\sum_{i=1}^n{f(x_i)}}\right) }[/math]

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
- Minkowski distance
- Quasi-arithmetic mean – another name for the generalized f-mean mentioned above
- Root mean square

## Notes

- ↑
^{1.0}^{1.1}Sýkora, Stanislav (2009).*Mathematical means and averages: basic properties*.**3**. Stan’s Library: Castano Primo, Italy. doi:10.3247/SL3Math09.001. - ↑
^{2.0}^{2.1}^{2.2}P. S. Bullen:*Handbook of Means and Their Inequalities*. Dordrecht, Netherlands: Kluwer, 2003, pp. 175-177 - ↑ Weisstein, Eric W.. "Power Mean". http://mathworld.wolfram.com/PowerMean.html. (retrieved 2019-08-17)
- ↑ Thompson, Sylvanus P. (1965).
*Calculus Made Easy*. Macmillan International Higher Education. p. 185. ISBN 9781349004874. https://books.google.com/books?id=6VJdDwAAQBAJ&pg=PA185. Retrieved 5 July 2020. - ↑ Jones, Alan R. (2018).
*Probability, Statistics and Other Frightening Stuff*. Routledge. p. 48. ISBN 9781351661386. https://books.google.com/books?id=OvtsDwAAQBAJ&pg=PA48. Retrieved 5 July 2020.

## 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

## External links

Original source: https://en.wikipedia.org/wiki/Generalized mean.
Read more |