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| {{Unreferenced|date=November 2009}}
| | I like my hobby Model Aircraft Hobbies. <br>I try to learn Vietnamese in my spare time.<br><br>Here is my webpage; [http://Www.bestsongsabout.com/country-songs-for-mothers/ Fifa 15 Coin Generator] |
| In mathematics, '''generalized means''' are a family of functions for aggregating sets of numbers, that include as special cases the [[arithmetic mean|arithmetic]], [[geometric mean|geometric]], and [[harmonic mean|harmonic]] [[mean]]s. The generalized mean is also known as '''power mean''' or '''Hölder mean''' (named after [[Otto Hölder]]).
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| ==Definition==
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| If ''p'' is a non-zero [[real number]], we can define the '''generalized mean''' or '''power mean''' with exponent ''p'' of the positive real numbers <math>x_1,\dots,x_n</math> as:
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| :<math>M_p(x_1,\dots,x_n) = \left( \frac{1}{n} \sum_{i=1}^n x_i^p \right)^{\frac{1}{p}}</math>
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| While for ''p'' = 0 we assume that it's equal to the geometric mean (which is, in fact, the limit of means with exponents approaching zero, as proved below for the general case):
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| :<math>M_0(x_1, \dots, x_n) = \sqrt[n]{\prod_{i=1}^n x_i}</math>
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| Furthermore, for a sequence of positive weights ''w<sub>i</sub>'' with sum <math>\sum w_i = 1</math> we define the '''weighted power mean''' as:
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| :<math>\begin{align}
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| M_p(x_1,\dots,x_n) &= \left(\sum_{i=1}^n w_i x_i^p \right)^{\frac{1}{p}} \\
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| M_0(x_1,\dots,x_n) &= \prod_{i=1}^n x_i^{w_i}
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| \end{align}</math>
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| The unweighted means correspond to setting all ''w<sub>i</sub>'' = 1/''n''. For exponents equal to positive or negative infinity the means are maximum and minimum, respectively, regardless of weights (and they are actually the limit points for exponents approaching the respective extremes, as proved below):
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| :<math>\begin{align}
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| M_{ \infty}(x_1, \dots, x_n) &= \max(x_1, \dots, x_n) \\
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| M_{-\infty}(x_1, \dots, x_n) &= \min(x_1, \dots, x_n)
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| \end{align}</math>
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| :{| class="toccolours collapsible collapsed" width="90%" style="text-align:left"
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| !Proof of <math>\textstyle \lim_{p \to 0} M_p = M_0</math> (geometric mean)
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| |-
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| We can rewrite the definition of ''M<sub>p</sub>'' using the [[exponential function]]
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| :<math>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>
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| In the limit ''p'' → 0, we can apply [[L'Hôpital's rule]] to the exponential component,
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| :<math>\lim_{p \to 0} \frac{\ln{\left(\sum_{i=1}^n w_ix_{i}^p \right)}}{p} = \lim_{p \to 0} \frac{\sum_{i=1}^n w_i x_i^p \ln{x_i}}{\sum_{i=1}^n w_i x_i^p} = \sum_{i=1}^n w_i \ln{x_i} = \ln{\left(\prod_{i=1}^n x_i^{w_i} \right)}</math>
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| By the continuity of the exponential function, we can substitute back into the above relation to obtain
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| :<math>\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>
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| as desired.
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| |}
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| :{| class="toccolours collapsible collapsed" width="90%" style="text-align:left"
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| !Proof of <math>\textstyle \lim_{p \to \infty} M_p = M_\infty</math> and <math>\textstyle \lim_{p \to -\infty} M_p = M_{-\infty}</math>
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| |-
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| Assume (possibly after relabeling and combining terms together) that <math>x_1 \geq \dots \geq x_n</math>. Then
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| :<math>\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).</math>
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| The formula for <math>M_{-\infty}</math> follows from <math>M_{-\infty} (x_1,\dots,x_n) = \frac{1}{M_\infty (1/x_1,\dots,1/x_n)}.</math>
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| |}
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| ==Properties==
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| <!--
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| The case ''t'' = 1 yields the [[arithmetic mean]] and the case ''t'' = −1 yields the [[harmonic mean]]. The case ''t'' = 2 yields the [[root mean square]]. As ''t'' approaches 0, the [[Limit (mathematics)|limit]] of ''M''(''t'') is the [[geometric mean]] of the given numbers, and so it makes sense to ''define'' ''M''(0) to be the geometric mean. Furthermore, as ''t'' approaches ∞, ''M''(''t'') approaches the maximum of the given numbers, and as ''t'' approaches −∞, ''M''(''t'') approaches the minimum of the given numbers.
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| -->
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| * Like most [[Mean#Properties|mean]]s, the generalized mean is a [[homogeneous function]] of its arguments ''x''<sub>1</sub>, ..., ''x<sub>n</sub>''. That is, if ''b'' is a positive real number, then the generalized mean with exponent ''p'' of the numbers <math>b\cdot x_1,\dots, b\cdot x_n</math> is equal to ''b'' times the generalized mean of the numbers ''x''<sub>1</sub>, …, ''x<sub>n</sub>''.
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| * Like the [[quasi-arithmetic mean]]s, the computation of the mean can be split into computations of equal sized sub-blocks.
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| :: <math>M_p(x_1, \dots, x_{n \cdot k}) = M_p(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}))</math>
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| === Generalized mean inequality ===
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| In general,
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| : if ''p'' < ''q'', then <math>M_p(x_1,\dots,x_n) \le M_q(x_1,\dots,x_n)</math>
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| and the two means are equal if and only if ''x''<sub>1</sub> = ''x''<sub>2</sub> = ... = ''x<sub>n</sub>''.
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| The inequality is true for real values if ''p'' and ''q'', as well as positive and negative infinity values.
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| It follows from the fact that, for all real ''p'',
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| : <math>\frac{\partial}{\partial p}M_p(x_1, \dots, x_n) \geq 0</math>
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| which can be proved using [[Jensen's inequality]].
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| 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]].
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| == Special cases ==
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| [[Image:MathematicalMeans.svg|thumb|right|A visual depiction of some of the specified cases for ''n=2''.]]
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| {| style="border:none; padding:20px"
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| |<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>
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| |style="padding:0px 0px 0px 20px"| [[minimum]]
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| |-
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| |<math>M_{-1}(x_1,\dots,x_n) = \frac{n}{\frac{1}{x_1}+\dots+\frac{1}{x_n}}</math>
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| |style="padding:0px 0px 0px 20px"| [[harmonic mean]]
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| |-
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| |<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>
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| |style="padding:0px 0px 0px 20px"| [[geometric mean]]
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| |-
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| |<math>M_1(x_1,\dots,x_n) = \frac{x_1 + \dots + x_n}{n}</math>
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| |style="padding:0px 0px 0px 20px"| [[arithmetic mean]]
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| |-
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| |<math>M_2(x_1,\dots,x_n) = \sqrt{\frac{x_1^2 + \dots + x_n^2}{n}}</math>
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| |style="padding:0px 0px 0px 20px"| quadratic mean, a.k.a. [[root mean square]]
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| |-
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| |<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>
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| |style="padding:0px 0px 0px 20px"| [[maximum]]
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| |}
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| ==Proof of power means inequality==
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| We will prove weighted power means inequality, for the purpose of the proof we will assume the following without loss of generality:
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| :<math>\begin{align}
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| w_i \in [0; 1] \\
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| \sum_{i=1}^nw_i = 1
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| \end{align}</math>
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| Proof for unweighted power means is easily obtained by substituting ''w<sub>i</sub>'' = 1/''n''.
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| ===Equivalence of inequalities between means of opposite signs===
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| Suppose an average between power means with exponents ''p'' and ''q'' holds:
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| :<math>\sqrt[p]{\sum_{i=1}^nw_ix_i^p}\geq \sqrt[q]{\sum_{i=1}^nw_ix_i^q}</math>
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| applying this, then:
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| :<math>\sqrt[p]{\sum_{i=1}^n\frac{w_i}{x_i^p}}\leq \sqrt[q]{\sum_{i=1}^n\frac{w_i}{x_i^q}}</math>
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| We raise both sides to the power of −1 (strictly decreasing function in positive reals):
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| :<math>\sqrt[-p]{\sum_{i=1}^nw_ix_i^{-p}}=\sqrt[p]{\frac{1}{\sum_{i=1}^nw_i\frac{1}{x_i^p}}}\geq \sqrt[q]{\frac{1}{\sum_{i=1}^nw_i\frac{1}{x_i^q}}}=\sqrt[-q]{\sum_{i=1}^nw_ix_i^{-q}}</math>
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| 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.
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| ===Geometric mean===
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| For any ''q'' the inequality between mean with exponent ''q'' and geometric mean can be transformed in the following way:
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| :<math>\begin{align}
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| \prod_{i=1}^nx_i^{w_i} &\leq \sqrt[q]{\sum_{i=1}^nw_ix_i^q} \\
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| \sqrt[q]{\sum_{i=1}^nw_ix_i^q} &\leq \prod_{i=1}^nx_i^{w_i}
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| \end{align}</math>
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| (the first inequality is to be proven for positive ''q'', and the latter otherwise)
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| We raise both sides to the power of ''q'':
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| :<math>\prod_{i=1}^nx_i^{w_i\cdot q} \leq \sum_{i=1}^nw_ix_i^q</math>
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| in both cases we get the inequality between weighted arithmetic and geometric means for the sequence <math>x_i^q</math>, which can be proved by [[Jensen's inequality]], making use of the fact the [[logarithm]]ic function is concave:
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| :<math>\begin{align}
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| \sum_{i=1}^nw_i\log(x_i) &\leq \log\left( \sum_{i=1}^nw_ix_i \right) \\
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| \log \left(\prod_{i=1}^nx_i^{w_i} \right) &\leq \log\left( \sum_{i=1}^nw_ix_i \right)
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| \end{align}</math>
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| 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
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| :<math>\prod_{i=1}^nx_i^{w_i} \leq \sum_{i=1}^nw_ix_i</math>
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| Thus for any positive ''q'' it is true that:
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| :<math>\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}</math>
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| thus we have proved the inequality between geometric mean and any power mean.
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| ===Inequality between any two power means===
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| We are to prove that for any ''p'' < ''q'' the following inequality holds:
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| :<math>\sqrt[p]{\sum_{i=1}^nw_ix_i^p}\leq \sqrt[q]{\sum_{i=1}^nw_ix_i^q}</math>
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| if ''p'' is negative, and ''q'' is positive, the inequality is equivalent to the one proved above:
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| :<math>\sqrt[p]{\sum_{i=1}^nw_ix_i^p}\leq \prod_{i=1}^nx_i^{w_i} \leq\sqrt[q]{\sum_{i=1}^nw_ix_i^q}</math>
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| The proof for positive ''p'' and ''q'' is as follows: Define the following function: ''f'' : '''R'''<sub>+</sub> → '''R'''<sub>+</sub> <math>f(x)=x^{\frac{q}{p}}</math>. ''f'' is a power function, so it does have a second derivative:
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| : <math>f''(x) = \left(\frac{q}{p} \right) \left( \frac{q}{p}-1 \right)x^{\frac{q}{p}-2}</math>
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| which is strictly positive within the domain of ''f'', since ''q'' > ''p'', so we know ''f'' is convex.
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| Using this, and the Jensen's inequality we get:
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| :<math>\begin{align}
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| f \left( \sum_{i=1}^nw_ix_i^p \right) &\leq \sum_{i=1}^nw_if(x_i^p) \\
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| \sqrt[\frac{p}{q}]{\sum_{i=1}^nw_ix_i^p} &\leq \sum_{i=1}^nw_ix_i^q
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| \end{align}</math>
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| 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:
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| :<math>\sqrt[p]{\sum_{i=1}^nw_ix_i^p}\leq\sqrt[q]{\sum_{i=1}^nw_ix_i^q}</math>
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| Using the previously shown equivalence we can prove the inequality for negative ''p'' and ''q'' by substituting them with, respectively, −''q'' and −''p'', QED.
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| == Generalized ''f''-mean ==
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| {{Main|Generalized f-mean|l1=Generalized ƒ-mean}}
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| The power mean could be generalized further to the [[generalized f-mean|generalized ''f''-mean]]:
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| :<math> M_f(x_1,\dots,x_n) = f^{-1} \left({\frac{1}{n}\cdot\sum_{i=1}^n{f(x_i)}}\right) </math>
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| Which covers the geometric mean without using a limit with ''f''(''x'') = ''log''(''x''). The power mean is obtained for ''f''(''x'') = ''x<sup>p</sup>''.
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| == Applications ==
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| ===Signal processing===
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| 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 [[lowpass|moving arithmetic mean]] called <code>smooth</code> you can implement a moving power mean according to the following [[Haskell (programming language)|Haskell]] code.
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| <source lang="haskell">
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| powerSmooth :: Floating a => ([a] -> [a]) -> a -> [a] -> [a]
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| powerSmooth smooth p = map (** recip p) . smooth . map (**p)
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| </source>
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| * For big ''p'' it can serve an [[envelope detector]] on a [[rectifier|rectified]] signal.
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| * For small ''p'' it can serve an [[Baseline (spectrometry)|baseline detector]] on a [[mass spectrum]].
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| ==See also==
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| * [[Arithmetic mean]]
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| * [[Arithmetic-geometric mean]]
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| * [[Average]]
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| * [[Geometric mean]]
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| * [[Harmonic mean]]
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| * [[Heronian mean]]
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| * [[Inequality of arithmetic and geometric means]]
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| * [[Lehmer mean]] – also a mean related to [[Power (mathematics)|powers]]
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| * [[Root mean square]]
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| ==External links==
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| *[http://mathworld.wolfram.com/PowerMean.html Power mean at MathWorld]
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| *[http://people.revoledu.com/kardi/tutorial/BasicMath/Average/Generalized%20mean.html Examples of Generalized Mean]
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| *A [http://planetmath.org/?op=getobj&from=objects&name=ProofOfGeneralMeansInequality proof of the Generalized Mean] on [[PlanetMath]]
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| [[Category:Means]]
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| [[Category:Inequalities]]
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| [[Category:Articles with example Haskell code]]
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