What is Rearrangement inequality?

In mathematics, the rearrangement inequality states that

then the lower bound is attained only for the permutation which reverses the order, that is, σ = n − i + 1 {\displaystyle \sigma =n-i+1} for all i = 1 , … , n , {\displaystyle i=1,\ldots ,n,} and the upper bound is attained only for the identity, that is, σ = i {\displaystyle \sigma =i} for all i = 1 , … , n . {\displaystyle i=1,\ldots ,n.}

Note that the rearrangement inequality makes no assumptions on the signs of the real numbers.