Rearrangement inequality

 In mathematics, the rearrangement inequality^[1] states that

$${\displaystyle x_{n}y_{1}+\cdots +x_{1}y_{n}\leqslant x_{\sigma (1)}y_{1}+\cdots +x_{\sigma (n)}y_{n}\leq x_{1}y_{1}+\cdots +x_{n}y_{n}}$$

for every choice of real numbers

$$ {\displaystyle x_{1}\leqslant \cdots \leqslant x_{n}\quad {\text{and}}\quad y_{1}\leqslant \cdots \leq y_{n}}$$

and every permutation $ {\displaystyle x_{\sigma (1)},\dots ,x_{\sigma (n)}}$ of $x_1,..,x_n.$

If the numbers are different, meaning that $$ {\displaystyle x_{1}<\cdots <x_{n}\quad {\text{and}}\quad y_{1}<\cdots <y_{n},}$$

Source. Wikipedia

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