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

<math>a_1 \geq a_2 \geq \cdots \geq a_n</math>

and

<math>b_1 \geq b_2 \geq \cdots \geq b_n,</math>

then

<math>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).</math>

Similarly, if

<math>a_1 \geq a_2 \geq \cdots \geq a_n</math>

and

<math>b_1 \leq b_2 \leq \cdots \leq b_n,</math>

then

<math>n \sum_{k=1}^n a_kb_k \leq \left(\sum_{k=1}^n a_k\right)\left(\sum_{k=1}^n b_k\right).</math>

Chebyshev's sum inequality follows from the rearrangement inequality.

There is also a continuous version of Chebyshev's inequality:

If f and g are real-valued, integrable functions over [0,1], both increasing or both decreasing, then