Rearrangement Inequality

In mathematics, the rearrangement inequality states that

x_ny_1 + cdots + x_1y_n
le x_{sigma (1)}y_1 + cdots + x_{sigma (n)}y_n
le x_1y_1 + cdots + x_ny_n

for every choice of real numbers

and every permutation

of x1, . . ., xn. If the numbers are different, meaning that

then the lower bound is attained only for the permutation which reverses the order, i.e. σ(i) = ni + 1 for all i = 1, ..., n, and the upper bound is attained only for the identity, i.e. σ(i) = i for all i = 1, ..., n.

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

Read more about Rearrangement InequalityApplications, Proof

Other articles related to "rearrangement inequality":

Rearrangement Inequality - Proof
... We will now prove by contradiction, that σ has to be the identity (then we are done) ... Assume that σ is not the identity ...

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