The power mean inequality

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

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

This formula yields nonsense if $r=0$, but there is a natural way to define $P_0$ too: it is simply defined to be the geometric mean¹:

$$P_0 := \sqrt[n]{x_1 x_2 \dots x_n}.$$

One also defines

$$P_\infty = \max\{x_1,\dots,x_n\}$$

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

$$P_{-\infty}=\min\{x_1,\dots,x_n\}.$$

Here are some examples:

$$P_1 = \frac{x_1+\dots+x_n}{n}$$

is the arithmetic mean,

$$P_2 = \sqrt{\frac{x_1^2+\dots+x_n^2}{n}}$$

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

$$P_{-1} = \frac{n}{\frac{1}{x_1} + \dots + \frac{1}{x_n}}$$

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 $x_i$ 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:

• $P_1 \ge P_0$ (the AM-GM inequality).
• $P_0 \ge P_{-1}$ (the GM-HM inequality -- HM is for ``harmonic mean'').
• $P_1 \ge P_{-1}$ (the AM-HM inequality).