Chebyshev's sum inequality

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For the similarly named inequality in probability theory, see Chebyshev's inequality.

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



Similarly, if





Consider the sum

The two sequences are non-increasing, therefore aj − ak and bj − bk have the same sign for any jk. Hence S ≥ 0.

Opening the brackets, we deduce:


An alternative proof is simply obtained with the rearrangement inequality.

Continuous version[edit]

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.


  1. ^ Hardy, G. H.; Littlewood, J. E.; Pólya, G. (1988). Inequalities. Cambridge Mathematical Library. Cambridge: Cambridge University Press. ISBN 0-521-35880-9. MR 0944909.