Chebyshev's sum inequality

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

and

then

In words, if we are given two sequences that are both non-increasing or non-decreasing, then the product of their averages is less than the average of their (termwise) product.

Similarly, if

and

then

Proof

Consider the sum

The two sequences are non-increasing, therefore and have the same sign for any . Hence .

Opening the brackets, we deduce:

hence

An alternative proof is simply obtained with the rearrangement inequality, writing that

Continuous version

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

If f and g are real-valued, integrable functions over [a, b], both non-increasing or both non-decreasing, then

with the inequality reversed if one is non-increasing and the other is non-decreasing.

See also

• Hardy–Littlewood inequality
• Rearrangement inequality

Notes