The power mean inequality

Fix $x_1,\dots,x_n \ge 0$. For $r \not=0$ (assume r>0 if some xi are zero), the r-th power mean Pr of $x_1,\dots,x_n$is defined to be the r-th root of the average of the r-th powers of $x_1,\dots,x_n$:

\begin{displaymath}P_r := \left( \frac{x_1^r + \dots + x_n^r}{n} \right)^{1/r}.\end{displaymath}

This formula yields nonsense if r=0, but there is a natural way to define P0 too: it is simply defined to be the geometric mean1:

\begin{displaymath}P_0 := \sqrt[n]{x_1 x_2 \dots x_n}.\end{displaymath}

One also defines

\begin{displaymath}P_\infty = \max\{x_1,\dots,x_n\}\end{displaymath}

since when r is very large, Pr is a good approximation to the largest of $x_1,\dots,x_n$. For a similar reason one uses the notation

\begin{displaymath}P_{-\infty}=\min\{x_1,\dots,x_n\}.\end{displaymath}

Here are some examples:

\begin{displaymath}P_1 = \frac{x_1+\dots+x_n}{n}\end{displaymath}

is the arithmetic mean,

\begin{displaymath}P_2 = \sqrt{\frac{x_1^2+\dots+x_n^2}{n}}\end{displaymath}

is sometimes called the root mean square. For $x_1,\dots,x_n>0$,

\begin{displaymath}P_{-1} = \frac{n}{\frac{1}{x_1} + \dots + \frac{1}{x_n}}\end{displaymath}

is called the harmonic mean.


Power mean inequality:
Let $x_1,\dots,x_n \ge 0$. Suppose r>s (and $s \ge 0$ if any of the xi are zero). Then $P_r \ge P_s$, with equality if and only if $x_1=x_2=\dots=x_n$.


The power mean inequality holds even if $r=\infty$ or $s=-\infty$, provided that we use the definitions of $P_\infty$ and $P_{-\infty}$above, and the convention that $\infty > r > -\infty$ for all numbers r.

Here are some special cases of the power mean inequality: