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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).
Definition
If p 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 p of these positive real numbers is
M_p(x_1,\dots,x_n) = \left( \frac{1}{n} \sum_{i=1}^n x_i^p \right)^{{1}/{p}} .
(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):
M_0(x_1, \dots, x_n) = \left(\prod_{i=1}^n x_i\right)^{1/n} .
Furthermore, for a sequence of positive weights wi we define the weighted power mean as

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