Questions tagged [rearrangement-inequality]

The rearrangement inequality:

For $a_1\le\dots\le a_n$, $b_1\le\dots\le b_n$, and any permutation $\sigma\in S_n$, $$\sum_{i=1}^na_ib_{n-i+1}\le\sum_{i=1}^na_ib_{\sigma(i)}\leq\sum_{i=1}^na_ib_i.$$

Chebyshov inequality:

For any $a_1\le\dots\le a_n$ and $b_1\le\dots\le b_n$, $$n\sum_{i=1}^na_ib_{n-i+1}\le\left(\sum_{i=1}^na_i\right)\left(\sum_{i=1}^nb_i\right)\le n\sum_{i=1}^na_ib_i.$$