Rearrangement inequality

Let

$x_1 \leq \cdots \leq x_n\quad \mbox{and}\quad y_1 \leq \cdots \leq y_n$

be real numbers and

$x_{\sigma (1)}, \dots ,x_{\sigma (n)}$

be any permutation of $x_1, \dots , x_n$ . Then the rearrangement inequality states that

$x_1y_1 + \cdots + x_ny_n \geq x_{\sigma (1)}y_1 + \cdots + x_{\sigma (n)}y_n \geq x_ny_1 + \cdots + x_1y_n.$

The rearrangement inequality can be proved by induction. Many famous inequalities can be proved by the rearrangement inequality, such as the arithmetic mean - geometric mean inequality, the Cauchy-Schwarz inequality, and Chebyshev's sum inequality.