In mathematics, Chebyshev's sum inequality, named after Pafnuty Chebyshev, states that if
Consider the sum
The two sequences are non-increasing, therefore aj − ak and bj − bk have the same sign for any j, k. Hence S ≥ 0.
Opening the brackets, we deduce:
An alternative proof is simply obtained with the rearrangement inequality, writing that
There is also a continuous version of Chebyshev's sum inequality:
If f and g are real-valued, integrable functions over [0,1], both non-increasing or both non-decreasing, then
with the inequality reversed if one is non-increasing and the other is non-decreasing.
- ^ Hardy, G. H.; Littlewood, J. E.; Pólya, G. (1988). Inequalities. Cambridge Mathematical Library. Cambridge: Cambridge University Press. ISBN 0-521-35880-9. MR 0944909.