Chebyshev's sum inequality

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

and

then

Similarly, if

and

then

[1]

Proof

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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:

 

hence

 

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

 

Continuous version

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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

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Notes

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  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.