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((Short description|N-th root of the arithmetic mean of the given numbers raised to the power n))
((More citations needed|date=June 2020))

[[File:Generalized means of 1, x.svg|400px|thumb|right|Plot of several generalized means <math>M_p(1, x)</math>.]]

In [[mathematics]], '''generalized means''' (or '''power mean''' or '''Hölder mean''' from [[Otto Hölder]])<ref name=sykora/> are a family of functions for aggregating sets of numbers. These include as special cases the [[Pythagorean means]] ([[arithmetic mean|arithmetic]], [[geometric mean|geometric]], and [[harmonic mean|harmonic]] [[mean]]s).

==Definition==
If ((mvar|p)) is a non-zero [[real number]], and <math>x_1, \dots, x_n</math> are positive real numbers, then the '''generalized mean''' or '''power mean''' with exponent ((mvar|p)) of these positive real numbers is<ref name="Bullen1"/><ref name = "dC2016">((cite journal|last=de Carvalho|first=Miguel|title=Mean, what do you Mean?|journal=[[The American Statistician]]|year=2016|volume=70|issue=3|pages=764‒776|doi=10.1080/00031305.2016.1148632|url=https://zenodo.org/record/895400|hdl=20.500.11820/fd7a8991-69a4-4fe5-876f-abcd2957a88c|hdl-access=free))</ref>

<math display=block>M_p(x_1,\dots,x_n) = \left( \frac{1}{n} \sum_{i=1}^n x_i^p \right)^((1}/{p)) .</math>

(See [[Norm (mathematics)#p-norm|((mvar|p))-norm]]). For ((math|1=''p'' = 0)) we set it equal to the geometric mean (which is the limit of means with exponents approaching zero, as proved below):

<math display="block">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 ((mvar|w<sub>i</sub>)) we define the '''weighted power mean''' as<ref name="Bullen1"/>
<math display=block>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 ((math|1=''p'' = 0)), it is equal to the [[weighted geometric mean]]:

<math display=block>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 ((math|1=''w<sub>i</sub>'' = 1/n)).

== Special cases ==
A few particular values of ((mvar|p)) yield special cases with their own names:<ref name="mw">((MathWorld|title=Power Mean|urlname=PowerMean)) (retrieved 2019-08-17)</ref>
;[[minimum]] :<math>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>
;[[Image:MathematicalMeans.svg|thumb|A visual depiction of some of the specified cases for ((math|1=''n'' = 2)) with ((math|1=''a'' = ''x''((sub|1)) = ''M''((sub|∞)))) and ((math|1=''b'' = ''x''((sub|2)) = ''M''((sub|−∞)))): ((legend|magenta|harmonic mean, ((math|''H'' ((=)) ''M''((sub|−1))(''a'', ''b''))),)) ((legend|blue|geometric mean, ((math|''G'' ((=)) ''M''((sub|0))(''a'', ''b''))))) ((legend|red|arithmetic mean, ((math|''A'' ((=)) ''M''((sub|1))(''a'', ''b''))))) ((legend|lime|quadratic mean, ((math|''Q'' ((=)) ''M''((sub|2))(''a'', ''b'')))))]][[harmonic mean]] :<math>M_{-1}(x_1,\dots,x_n) = \frac{n}{\frac{1}{x_1}+\dots+\frac{1}{x_n))</math>
;[[geometric mean]] <math>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>M_1(x_1,\dots,x_n) = \frac{x_1 + \dots + x_n}{n}</math>
;[[root mean square]]((anchor|Quadratic))<br/>or quadratic mean<ref>((cite book |last1=Thompson |first1=Sylvanus P. |title=Calculus Made Easy |date=1965 |publisher=Macmillan International Higher Education |isbn=9781349004874 |page=185 |url=https://books.google.com/books?id=6VJdDwAAQBAJ&pg=PA185 |access-date=5 July 2020))</ref><ref>((cite book |last1=Jones |first1=Alan R. |title=Probability, Statistics and Other Frightening Stuff |date=2018 |publisher=Routledge |isbn=9781351661386 |page=48 |url=https://books.google.com/books?id=OvtsDwAAQBAJ&pg=PA48 |access-date=5 July 2020))</ref> :<math>M_2(x_1,\dots,x_n) = \sqrt{\frac{x_1^2 + \dots + x_n^2}{n))</math>
;[[cubic mean]] :<math>M_3(x_1,\dots,x_n) = \sqrt[3]{\frac{x_1^3 + \dots + x_n^3}{n))</math>
;[[maximum]] :<math>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>

((Math proof|title=Proof of <math display="inline"> \lim_{p \to 0} M_p = M_0 </math> (geometric mean)|proof=For the purpose of the proof, we will assume without loss of generality that 
<math display="block"> w_i \in [0,1] </math>
and 
<math display="block"> \sum_{i=1}^n w_i = 1. </math>

We can rewrite the definition of <math>M_p</math> using the exponential function as

<math display=block>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 ((math|''p'' → 0)), we can apply [[L'Hôpital's rule]] to the argument of the exponential function. We assume that ((mvar|p)) ∈ ℝ but ((mvar|p)) ≠ 0, and that the sum of ((mvar|w<sub>i</sub>)) is equal to 1 (without loss in generality);<ref>((Cite book |title=Handbook of Means and Their Inequalities (Mathematics and Its Applications)))</ref> Differentiating the numerator and denominator with respect to ((mvar|p)), we have
<math display=block>\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} \\
 &= \frac{\sum_{i=1}^n w_i \ln{x_i)){\sum_{j=1}^n w_j} \\
 &= \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 display=block>\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.<ref name="Bullen1">P. S. Bullen: ''Handbook of Means and Their Inequalities''. Dordrecht, Netherlands: Kluwer, 2003, pp. 175-177</ref>))

((Proof|title= Proof of <math display="inline">\lim_{p \to \infty} M_p = M_\infty</math> and <math display="inline">\lim_{p \to -\infty} M_p = M_{-\infty}</math> |proof=
Assume (possibly after relabeling and combining terms together) that <math>x_1 \geq \dots \geq x_n</math>. Then

<math display=block>\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>M_{-\infty}</math> follows from 
<math display="block">M_{-\infty} (x_1,\dots,x_n) = \frac{1}{M_\infty (1/x_1,\dots,1/x_n)} = x_n.</math>
))


==Properties==

Let <math>x_1, \dots, x_n</math> be a sequence of positive real numbers, then the following properties hold:<ref name=sykora>((cite journal|last=Sýkora|first=Stanislav|year=2009|title=Mathematical means and averages: basic properties|journal=Stan's Library |publisher=Stan’s Library: Castano Primo, Italy|volume=3|issue=Volume III |doi=10.3247/SL3Math09.001 ))</ref>

#<math>\min(x_1, \dots, x_n) \le M_p(x_1, \dots, x_n) \le \max(x_1, \dots, x_n)</math>.<!--
-->((block indent|left=1|text= Each generalized mean always lies between the smallest and largest of the ((mvar|x)) values.))
#<math>M_p(x_1, \dots, x_n) = M_p(P(x_1, \dots, x_n))</math>, where <math>P</math> is a permutation operator.<!--
-->((block indent|left=1|text= Each generalized mean is a symmetric function of its arguments; permuting the arguments of a generalized mean does not change its value.))
#<math>M_p(b x_1, \dots, b x_n) = b \cdot M_p(x_1, \dots, x_n)</math>.<!--
-->((block indent|left=1|text= Like most [[Mean#Properties|mean]]s, the generalized mean is a [[homogeneous function]] of its arguments ((math|''x''<sub>1</sub>, ..., ''x<sub>n</sub>'')). That is, if ((mvar|b)) is a positive real number, then the generalized mean with exponent ((mvar|p)) of the numbers <math>b\cdot x_1,\dots, b\cdot x_n</math> is equal to ((mvar|b)) times the generalized mean of the numbers ((math|''x''<sub>1</sub>, ..., ''x<sub>n</sub>'')).))
#<math>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>.<!--
-->((block indent|left=1|text= Like the [[quasi-arithmetic mean]]s, 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 ===
((QM_AM_GM_HM_inequality_visual_proof.svg))
In general, if ((math|''p''&thinsp;<&thinsp;''q'')), then
<math display=block>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 ((math|1= ''x''<sub>1</sub>&thinsp;=&thinsp;''x''<sub>2</sub>&thinsp;=&thinsp;...&thinsp;=&thinsp;''x<sub>n</sub>'')).

The inequality is true for real values of ((mvar|p)) and ((mvar|q)), as well as positive and negative infinity values.

It follows from the fact that, for all real ((mvar|p)),
<math display=block>\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 ((mvar|p)) in ((math|{−1, 0, 1}<nowiki/>)), the generalized mean inequality implies the [[Pythagorean means]] inequality as well as the [[inequality of arithmetic and geometric means]].

==Proof of the weighted inequality==
We will prove the weighted power mean inequality. For the purpose of the proof we will assume the following without loss of generality:
<math display=block>\begin{align}
  w_i \in [0, 1] \\
  \sum_{i=1}^nw_i = 1
\end{align}</math>

The proof for unweighted power means can be easily obtained by substituting ((math|1= ''w<sub>i</sub>'' = 1/''n'')).

===Equivalence of inequalities between means of opposite signs===
Suppose an average between power means with exponents ((mvar|p)) and ((mvar|q)) holds:
<math display="block">\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 display="block">\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 display="block">\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 ((math|−''p'')) and ((math|−''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 ((math|''q'' > 0)) and non-negative weights summing to 1, the following inequality holds:
<math display="block">\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 display=block>\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 display=block>\prod_{i=1}^n x_i^{w_i} \leq \sum_{i=1}^n w_i x_i.</math>

Taking ((mvar|q))-th powers of the ((mvar|x<sub>i</sub>)) yields
<math display=block>\begin{align}
&\prod_{i=1}^n x_i^{q{\cdot}w_i} \leq \sum_{i=1}^n w_i x_i^q \\
&\prod_{i=1}^n x_i^{w_i} \leq \left(\sum_{i=1}^n w_i x_i^q\right)^{1/q}.\end{align}</math>

Thus, we are done for the inequality with positive ((mvar|q)); the case for negatives is identical but for the swapped signs in the last step:

<math display=block>\prod_{i=1}^n x_i^{-q{\cdot}w_i} \leq \sum_{i=1}^n w_i x_i^{-q}.</math>

Of course, taking each side to the power of a negative number ((math|-1/''q'')) swaps the direction of the inequality.

<math display=block>\prod_{i=1}^n x_i^{w_i} \geq \left(\sum_{i=1}^n w_i x_i^q\right)^{1/q}.</math>

===Inequality between any two power means===
We are to prove that for any ((math|''p'' < ''q'')) the following inequality holds:
<math display="block">\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 ((mvar|p)) is negative, and ((mvar|q)) is positive, the inequality is equivalent to the one proved above:
<math display="block">\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 ((mvar|p)) and ((mvar|q)) is as follows: Define the following function: ((math|''f'' : '''R'''<sub>+</sub> → '''R'''<sub>+</sub>)) <math>f(x)=x^{\frac{q}{p))</math>. ((mvar|f)) is a power function, so it does have a second derivative:
<math display="block">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 ((mvar|f)), since ((math|''q'' > ''p'')), so we know ((mvar|f)) is convex.

Using this, and the Jensen's inequality we get:
<math display="block">\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 ((math|1/''q'')) (an increasing function, since ((math|1/''q'')) is positive) we get the inequality which was to be proven:

<math display="block">\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 ((mvar|p)) and ((mvar|q)) by replacing them with ((mvar|&minus;q)) and ((mvar|&minus;p)), respectively.

== Generalized ''f''-mean ==
((Main|Generalized f-mean|l1=Generalized ((mvar|f))-mean))

The power mean could be generalized further to the [[generalized f-mean|generalized ((mvar|f))-mean]]:

<math display=block> 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 ((math|1= ''f''(''x'') ((=)) log(''x''))). The power mean is obtained for ((mvar|1= ''f''(''x'') ((=)) ''x<sup>p</sup>'')). Properties of these means are studied in de Carvalho (2016).<ref name = "dC2016"/>

== Applications ==

===Signal processing===
A power mean serves a non-linear [[moving average]] which is shifted towards small signal values for small ((mvar|p)) and emphasizes big signal values for big ((mvar|p)). Given an efficient implementation of a [[lowpass|moving arithmetic mean]] called <code>smooth</code> one can implement a moving power mean according to the following [[Haskell (programming language)|Haskell]] code.

<syntaxhighlight lang="haskell">
powerSmooth :: Floating a => ([a] -> [a]) -> a -> [a] -> [a]
powerSmooth smooth p = map (** recip p) . smooth . map (**p)
</syntaxhighlight>

* For big ((mvar|p)) it can serve as an [[envelope detector]] on a [[rectifier|rectified]] signal.
* For small ((mvar|p)) it can serve as a [[Baseline (spectrometry)|baseline detector]] on a [[mass spectrum]].

==See also==
((cols|colwidth=26em))
* [[Arithmetic–geometric mean]]
* [[Average]]
* [[Heronian mean]]
* [[Inequality of arithmetic and geometric means]]
* [[Lehmer mean]] &ndash; also a mean related to [[Power (mathematics)|powers]]
* [[Minkowski distance]]
* [[Quasi-arithmetic mean]] &ndash; another name for the [[generalized f-mean]] mentioned above
* [[Root mean square]]
((colend))

== Notes ==
((notelist))
((reflist|group=note))

== References ==
((reflist))

== Further reading ==
* ((cite book|first1=P. S. |last1=Bullen|title=Handbook of Means and Their Inequalities|location=Dordrecht, Netherlands|publisher=Kluwer|year=2003|chapter=Chapter III - The Power Means|pages=175–265))

==External links==
*[http://mathworld.wolfram.com/PowerMean.html Power mean at MathWorld]
*[http://people.revoledu.com/kardi/tutorial/BasicMath/Average/Generalized%20mean.html Examples of Generalized Mean]
*A [https://planetmath.org/ProofOfGeneralMeansInequality proof of the Generalized Mean] on [[PlanetMath]]

[[Category:Means]]
[[Category:Inequalities]]
[[Category:Articles with example Haskell code]]

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Module:Color contrast/colors (view source) (template editor protected)
Module:Disambiguation/templates (view source) (protected)
Module:Find sources (view source) (template editor protected)
Module:Find sources/config (view source) (template editor protected)
Module:Find sources/links (view source) (template editor protected)
Module:Find sources/templates/Find sources mainspace (view source) (template editor protected)
Module:Footnotes (view source) (template editor protected)
Module:Format link (view source) (template editor protected)
Module:Hatnote (view source) (template editor protected)
Module:Hatnote/styles.css (view source) (template editor protected)
Module:Hatnote list (view source) (template editor protected)
Module:Labelled list hatnote (view source) (template editor protected)
Module:Message box (view source) (protected)
Module:Message box/ambox.css (view source) (protected)
Module:Message box/configuration (view source) (protected)
Module:Namespace detect/config (view source) (protected)
Module:Namespace detect/data (view source) (protected)
Module:Pagetype (view source) (protected)
Module:Pagetype/config (view source) (protected)
Module:Pagetype/disambiguation (view source) (protected)
Module:Pagetype/setindex (view source) (protected)
Module:Pagetype/softredirect (view source) (protected)
Module:SDcat (view source) (protected)
Module:String (view source) (protected)
Module:TableTools (view source) (protected)
Module:Template wrapper (view source) (template editor protected)
Module:Unsubst (view source) (protected)
Module:Wikitext Parsing (view source) (protected)
Module:Yesno (view source) (protected)

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