power mean

generalized mean, Hölder mean, p-mean
#analysis

Definition

A power mean is a mean of form

M_p(a_1,a_2,...,a_n) \equiv \left(\frac{1}{n} \sum_{k=1}^n a_k^p\right)^{1/p}

where $p \in \mathbb{R}\setminus \{0\}$ , and $a_k \geq 0$

Notes

sometimes given as $p$ as an affinely extended real number (reals and positive/negative infinity), in which case, for positive and negative infinity, and 0, the limits are taken
special cases
- $M_{-\infty}= \lim_{p \to -\infty} M_p(a_1,a_2,...,a_n) = \min(a_1,a_2,...,a_n)$ , minimum
- $M_1$ , arithmetic mean
- etc...

References