A power mean is a type of mean.
Given positive real numbers \(a_1\), \(a_2\), …, \(a_n\), the \(p\)th power mean is obtained by taking the arithmetic mean of the \(p\)th powers of \(a_1\), …, \(a_n\), and then taking the \(p\)th root of this: \[\left(\frac{a_1^p+a_2^p+\cdots+a_n^p}{n}\right)^{\frac{1}{p}}\]
There are some familiar special cases:
\(p=1\) is the arithmetic mean
\(p=-1\) is the harmonic mean
\(p=2\) is the root mean square
(The arithmetic mean and root mean square also work even if some of the numbers are zero or negative.)
Different power means for the same \(a_1\), …, \(a_n\) satisfy the inequality: \[\text{if $p>q$, then the $p$th power mean $\ge$ the $q$th power mean}\] with equality if and only if all of the \(a_i\) are equal to each other.
