Generalized mean

In mathematics, generalized means are a family of functions for aggregating sets of numbers, that include as special cases the Pythagorean means (arithmetic, geometric, and harmonic means). The generalized mean is also known as power mean or Hölder mean (named after Otto Hölder).

If p is a non-zero real number, and ${\displaystyle x_{1},\dots ,x_{n}}$ are positive real numbers, then the generalized mean or power mean with exponent p of these positive real numbers is:

Note the relationship to the 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 w_i with sum ${\displaystyle \sum w_{i}=1}$ we define the weighted power mean as:

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