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Concept
Generalized mean
Summary
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).
If p is a non-zero real number, and are positive real numbers, then the generalized mean or power mean with exponent p of these positive real numbers is
(See p-norm). For p = 0 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 wi we define the weighted power mean as
and when p = 0, it is equal to the weighted geometric mean:
The unweighted means correspond to setting all wi = 1/n.
A few particular values of p yield special cases with their own names:
minimum
harmonic mean
geometric mean
arithmetic mean
root mean squareor quadratic mean
cubic mean
maximum
Let be a sequence of positive real numbers, then the following properties hold:
where is a permutation operator.
In general, if p < q, then
and the two means are equal if and only if x1 = x2 = ... = xn.
The inequality is true for real values of p and q, as well as positive and negative infinity values.
It follows from the fact that, for all real p,
which can be proved using Jensen's inequality.
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.
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:
applying this, then:
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.
Official source
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