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) = n − i + 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 Inequality:  Applications, Proof

Famous quotes containing the word inequality:

    “Nature is unfair? So much the better, inequality is the only bearable thing, the monotony of equality can only lead us to boredom.”
    —Francis Picabia (1878–1953)