Generalized Mean Inequality

In mathematics, generalized means (or power mean or Hölder mean from Otto Hölder) 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 is a non-zero real number, and $x_1, \dots, x_n$ are positive real numbers, then the generalized mean or power mean with exponent of these positive real numbers is:

$M_p(x_1,\dots,x_n) = \left( \frac \sum_^n x_i^p \right)^ .$

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

$M_0(x_1, \dots, x_n) = \left(\prod_^n x_i\right)^ .$

Furthermore, for a sequence of positive weights we define the weighted power mean as:
$M_p(x_1,\dots,x_n) = \left(\frac \right)^$
and when , it is equal to the weighted geometric mean:

$M_0(x_1,\dots,x_n) = \left(\prod_^n x_i^\right)^ .$

The unweighted means correspond to setting all .

Special cases

A few particular values of yield special cases with their own names: (retrieved 2019-08-17)
;minimum :$M_(x_1,\dots,x_n) = \lim_ M_p(x_1,\dots,x_n) = \min \$
;harmonic mean :$M_(x_1,\dots,x_n) = \frac$
;geometric mean :M_0(x_1,\dots,x_n) = \lim_ M_p(x_1,\dots,x_n) = \sqrt /math>
;arithmetic mean :$M_1(x_1,\dots,x_n) = \frac$
;root mean square
or quadratic mean :$M_2(x_1,\dots,x_n) = \sqrt$
;cubic mean :M_3(x_1,\dots,x_n) = \sqrt /math>
;maximum :$M_(x_1,\dots,x_n) = \lim_ M_p(x_1,\dots,x_n) = \max \$

Proof of $\lim_ M_p = M_0$ (geometric mean)
We can rewrite the definition of using the exponential function

$M_p(x_1,\dots,x_n) = \exp = \exp$

In the limit , we can apply L'Hôpital's rule to the argument of the exponential function. We assume that ∈ R but ≠ 0, and that the sum of is equal to 1 (without loss in generality); Differentiating the numerator and denominator with respect to , we have
$\begin \lim_ \frac &= \lim_ \frac \\ &= \lim_ \frac \\ &= \sum_^n \frac \\ &= \sum_^n w_i \ln \\ &= \ln \end$

By the continuity of the exponential function, we can substitute back into the above relation to obtain
$\lim_ M_p(x_1,\dots,x_n) = \exp = \prod_^n x_i^ = M_0(x_1,\dots,x_n)$
as desired.P. S. Bullen: ''Handbook of Means and Their Inequalities''. Dordrecht, Netherlands: Kluwer, 2003, pp. 175-177

Properties

Let $x_1, \dots, x_n$ be a sequence of positive real numbers, then the following properties hold:

#$\min(x_1, \dots, x_n) \le M_p(x_1, \dots, x_n) \le \max(x_1, \dots, x_n)$.
#$M_p(x_1, \dots, x_n) = M_p(P(x_1, \dots, x_n))$, where $P$ is a permutation operator.
#$M_p(b x_1, \dots, b x_n) = b \cdot M_p(x_1, \dots, x_n)$.
#M_p(x_1, \dots, x_) = M_p\left _p(x_1, \dots, x_), M_p(x_, \dots, x_), \dots, M_p(x_, \dots, x_)\right/math>.

Generalized mean inequality

In general, if , then
$M_p(x_1, \dots, x_n) \le M_q(x_1, \dots, x_n)$
and the two means are equal if and only if .

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

It follows from the fact that, for all real ,
$\fracM_p(x_1, \dots, x_n) \geq 0$
which can be proved using Jensen's inequality.

In particular, for in , 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:
$\begin w_i \in$, 1\\
\sum_^nw_i = 1
\end

Proof for unweighted power means is easily obtained by substituting .

Equivalence of inequalities between means of opposite signs

Suppose an average between power means with exponents and holds:
$\left(\sum_^n w_i x_i^p\right)^ \geq \left(\sum_^n w_i x_i^q\right)^$
applying this, then:
$\left(\sum_^n\frac\right)^ \geq \left(\sum_^n\frac\right)^$

We raise both sides to the power of −1 (strictly decreasing function in positive reals):
$\left(\sum_^nw_ix_i^\right)^ = \left(\frac\right)^ \leq \left(\frac\right)^ = \left(\sum_^nw_ix_i^\right)^$

We get the inequality for means with exponents and , 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 and non-negative weights summing to 1, the following inequality holds:
$\left(\sum_^n w_i x_i^\right)^ \leq \prod_^n x_i^ \leq \left(\sum_^n w_i x_i^q\right)^.$

The proof follows from Jensen's inequality, making use of the fact the logarithm is concave:
$\log \prod_^n x_i^ = \sum_^n w_i\log x_i \leq \log \sum_^n w_i x_i.$

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
$\prod_^n x_i^ \leq \sum_^n w_i x_i.$

Taking -th powers of the , we are done for the inequality with positive ; the case for negatives is identical.

Inequality between any two power means

We are to prove that for any the following inequality holds:
$\left(\sum_^n w_i x_i^p\right)^ \leq \left(\sum_^nw_ix_i^q\right)^$
if is negative, and is positive, the inequality is equivalent to the one proved above:
$\left(\sum_^nw_i x_i^p\right)^ \leq \prod_^n x_i^ \leq \left(\sum_^n w_i x_i^q\right)^$

The proof for positive and is as follows: Define the following function: $f(x)=x^$. is a power function, so it does have a second derivative:
$f''(x) = \left(\frac \right) \left( \frac-1 \right)x^$
which is strictly positive within the domain of , since , so we know is convex.

Using this, and the Jensen's inequality we get:
\begin
f \left( \sum_^nw_ix_i^p \right) &\leq \sum_^nw_if(x_i^p) \\ pt \left(\sum_^n w_i x_i^p\right)^ &\leq \sum_^nw_ix_i^q
\end
after raising both side to the power of (an increasing function, since is positive) we get the inequality which was to be proven:

$\left(\sum_^n w_i x_i^p\right)^ \leq \left(\sum_^n w_i x_i^q\right)^$

Using the previously shown equivalence we can prove the inequality for negative and by replacing them with and , respectively.

Generalized ''f''-mean

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

$M_f(x_1,\dots,x_n) = f^ \left(\right)$

This covers the geometric mean without using a limit with . The power mean is obtained for . Properties of these means are studied in de Carvalho (2016).

Applications

Signal processing

A power mean serves a non-linear moving average which is shifted towards small signal values for small and emphasizes big signal values for big . 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 -> -> powerSmooth smooth p = map (** recip p) . smooth . map (**p)

* For big it can serve as an envelope detector on a rectified signal.
* For small {{mvar, 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

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

Power mean at MathWorld
*
proof of the Generalized Mean
on PlanetMath

Means
Inequalities
Articles with example Haskell code