A generalized mean, also known as power mean or Hölder mean, is an abstraction of the Pythagorean means including arithmetic, geometric and harmonic means.

Definition

If t is a non-zero real number, we can define the generalized mean with exponent t of the positive real numbers a₁,...,a_n as

$$

M(t) = \left( \frac{1}{n} \sum_{i=1}^n a_{i}^t \right) ^ {\frac{1}{t}}

Properties

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 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.

The generalized mean is a homogeneous function of its arguments a₁,..., a_n. That is, if b is a positive real number, then the generalized mean with exponent t of the numbers ba₁,..., ba_n is equal to b times the generalized mean of the numbers a₁,..., a_n.

Generalized Mean Inequality

In general, if −∞ ≤ s < t ≤ ∞, then M(s) ≤ M(t) and the two means are equal if and only if a₁ = a₂ = ... = a_n. That follows from the fact that $\backslash frac\{\backslash partial\; M(t)\}\{\backslash partial\; t\}\backslash geq\; 0$ for −∞ ≤ t ≤ ∞, that can be proved using Jensen's inequality.

In particular, for t equal -1, 0, and 1, the generalized mean inequality implies the Pythagorean means inequality as well as the inequality of arithmetic and geometric means.

Generalized f-mean

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

$M\; =\; f^\{-1\}\backslash left(\{\backslash frac\{1\}\{n\}\backslash sum\_\{i=1\}^n\{f(x\_i)\}\}\backslash right)$

and again a suitable choice of an invertible f(x) will give the arithmetic mean with f(x) = x, the geometric mean with f(x) = log(x), the harmonic mean with f(x) = 1/x, and the generalized mean with exponent t with f(x) = x^t. But other functions could be used, such as f(x) = e^x.

See also

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• arithmetic mean
• geometric mean
• harmonic mean
• average
• root mean square

External links

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• Power mean at MathWorld
• Examples of Generalized Mean

Means | Inequalities

Wortelgemiddelde | Średnia uogólniona | Неравенство о средних | Generalized mean