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.