Chebyshev's sum inequality

Another article treats Chebyshev's inequality in probability theory.

In mathematics, Chebyshev's sum inequality, named after Pafnuty Chebyshev, states that if

$a_1 \geq a_2 \geq \cdots \geq a_n$

and

$b_1 \geq b_2 \geq \cdots \geq b_n,$

then

$n \sum_{k=1}^n a_kb_k \geq \left(\sum_{k=1}^n a_k\right)\left(\sum_{k=1}^n b_k\right).$

Chebyshev's sum inequality follows from the rearrangement inequality.

See also: Chebyshev's sum inequality, Chebyshev's inequality, Mathematics, Pafnuty Chebyshev, Probability theory, Rearrangement inequality