If Template:Mvar 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 Template:Mvar of these positive real numbers is<ref name="Bullen1"/><ref name = "dC2016">Template:Cite journal</ref>
(See [[Norm (mathematics)#p-norm|Template:Mvar-norm]]). For Template:Math 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 Template:Mvar 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 Template:Math, it is equal to the weighted geometric mean:
In the limit Template:Math, we can apply L'Hôpital's rule to the argument of the exponential function. We assume that <math>p \isin \mathbb{R}</math> but Template:Math, and that the sum of Template:Mvar is equal to 1 (without loss in generality);<ref>Template:Cite book</ref> Differentiating the numerator and denominator with respect to Template:Mvar, we have
<math display=block>\begin{align}
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>}}
Let <math>x_1, \dots, x_n</math> be a sequence of positive real numbers, then the following properties hold:<ref name=sykora>Template:Cite journal</ref>
The inequality is true for real values of Template:Mvar and Template:Mvar, as well as positive and negative infinity values.
It follows from the fact that, for all real Template:Mvar,
<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.
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 Template:Math.
Equivalence of inequalities between means of opposite signs
Suppose an average between power means with exponents Template:Mvar and Template:Mvar 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 Template:Math and Template:Math, 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.
For any Template:Math 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>
We are to prove that for any Template:Math 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 Template:Mvar is negative, and Template:Mvar 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 Template:Mvar and Template:Mvar is as follows: Define the following function: Template:Math <math>f(x)=x^{\frac{q}{p}}</math>. Template:Mvar 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 Template:Mvar, since Template:Math, so we know Template:Mvar is convex.
Using this, and the Jensen's inequality we get:
<math display="block">\begin{align}
\end{align}</math>
after raising both side to the power of Template:Math (an increasing function, since Template:Math is positive) we get the inequality which was to be proven:
This covers the geometric mean without using a limit with Template:Math. The power mean is obtained for Template:Mvar. Properties of these means are studied in de Carvalho (2016).<ref name = "dC2016"/>
A power mean serves a non-linear moving average which is shifted towards small signal values for small Template:Mvar and emphasizes big signal values for big Template:Mvar. Given an efficient implementation of a moving arithmetic mean called smooth one can implement a moving power mean according to the following Haskell code.
<syntaxhighlight lang="haskell">
powerSmooth :: Floating a => ([a] -> [a]) -> a -> [a] -> [a]
powerSmooth smooth p = map (** recip p) . smooth . map (**p)
</syntaxhighlight>
