Chebyshev's sum inequality


In mathematics, Chebyshev's sum inequality, named after Pafnuty Chebyshev, states that if
and
then
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:
whence
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 , both non-increasing or both non-decreasing, then
with the inequality reversed if one is non-increasing and the other is non-decreasing.