Generalized mean

are positive real numbers, then the generalized mean or power mean with exponent p of these positive real numbers is[2][3]

and when p = 0, it is equal to the weighted geometric mean:

The unweighted means correspond to setting all wi = 1.

A few particular values of p yield special cases with their own names:[4] For the purpose of the proof, we will assume without loss of generality that

In the limit p → 0, we can apply L'Hôpital's rule to the argument of the exponential function.

but p ≠ 0, and that the sum of wi is equal to 1 (without loss in generality);[7] Differentiating the numerator and denominator with respect to p, we have

By the continuity of the exponential function, we can substitute back into the above relation to obtain

[2] Assume (possibly after relabeling and combining terms together) that

be a sequence of positive real numbers, then the following properties hold:[1] In general, if p < q, then

We will prove the weighted power mean inequality.

For the purpose of the proof we will assume the following without loss of generality:

The proof for unweighted power means can be easily obtained by substituting wi = 1/n.

Suppose an average between power means with exponents p and q holds:

We raise both sides to the power of −1 (strictly decreasing function in positive reals):

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.

For any q > 0 and non-negative weights summing to 1, the following inequality holds:

The proof follows from Jensen's inequality, making use of the fact the logarithm is concave:

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

Taking q-th powers of the xi yields

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

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

if p is negative, and q is positive, the inequality is equivalent to the one proved above:

The proof for positive p and q is as follows: Define the following function: f : R+ → R+

which is strictly positive within the domain of f, since q > p, so we know f is convex.

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:

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

This covers the geometric mean without using a limit with f(x) = log(x).

The power mean is obtained for f(x) = xp.

Properties of these means are studied in de Carvalho (2016).

[3] 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.