# Chebyshev's sum inequality

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

${\displaystyle a_{1}\geq a_{2}\geq \cdots \geq a_{n}}$

and

${\displaystyle b_{1}\geq b_{2}\geq \cdots \geq b_{n},}$

then

${\displaystyle {1 \over n}\sum _{k=1}^{n}a_{k}\cdot b_{k}\geq \left({1 \over n}\sum _{k=1}^{n}a_{k}\right)\left({1 \over n}\sum _{k=1}^{n}b_{k}\right).}$

Similarly, if

${\displaystyle a_{1}\leq a_{2}\leq \cdots \leq a_{n}}$

and

${\displaystyle b_{1}\geq b_{2}\geq \cdots \geq b_{n},}$

then

${\displaystyle {1 \over n}\sum _{k=1}^{n}a_{k}b_{k}\leq \left({1 \over n}\sum _{k=1}^{n}a_{k}\right)\left({1 \over n}\sum _{k=1}^{n}b_{k}\right).}$[1]

## Proof

Consider the sum

${\displaystyle S=\sum _{j=1}^{n}\sum _{k=1}^{n}(a_{j}-a_{k})(b_{j}-b_{k}).}$

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:

${\displaystyle 0\leq 2n\sum _{j=1}^{n}a_{j}b_{j}-2\sum _{j=1}^{n}a_{j}\,\sum _{k=1}^{n}b_{k},}$

whence

${\displaystyle {\frac {1}{n}}\sum _{j=1}^{n}a_{j}b_{j}\geq \left({\frac {1}{n}}\sum _{j=1}^{n}a_{j}\right)\,\left({\frac {1}{n}}\sum _{k=1}^{n}b_{k}\right).}$

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

${\displaystyle \sum _{i=0}^{n-1}a_{i}\sum _{j=0}^{n-1}b_{i}=\sum _{i=0}^{n-1}\sum _{k=0}^{n-1}a_{i}b_{i+k\ mod\ n}\leq \sum _{i=0}^{n-1}\sum _{k=0}^{n-1}a_{k}b_{k}=n\sum _{k}a_{k}b_{k}.}$

## Continuous version

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

${\displaystyle \int _{0}^{1}f(x)g(x)\,dx\geq \int _{0}^{1}f(x)\,dx\int _{0}^{1}g(x)\,dx,}$

with the inequality reversed if one is non-increasing and the other is non-decreasing.

## Notes

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.