The power means inequality is a generalization of arithmetic-geometric means inequality.

If , the $r$ -mean (or $r$ -th power mean) of the nonnegative numbers $a_1,\ldots,a_n$ is defined as $$M^r(a_1,a_2,\ldots,a_n)= \left(\frac{1}{n}\displaystyle{\sum_{k=1}^n a_k^r}\right)^{1/r}$$

Given real numbers $x,y$ such that $xy\neq 0$ and $x<y$ , we have $$M^x \leq M^y$$ and the equality holds if and only if $a_1 = ... = a_n$ .

Additionally, if we define $M^0$ to be the geometric mean $(a_1a_2...a_n)^{1/n}$ , we have that the inequality above holds for arbitrary real numbers $x<y$ .

The mentioned inequality is a special case of this one, since $M^1$ is the arithmetic mean, $M^0$ is the geometric mean and $M^{-1}$ is the harmonic mean.

This inequality can be further generalized using weighted power means.