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