In mathematics , Chebyshev's sum inequality , named after Pafnuty Chebyshev , states that if
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≥
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{\displaystyle a_{1}\geq a_{2}\geq \cdots \geq a_{n}\quad }
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{\displaystyle \quad b_{1}\geq b_{2}\geq \cdots \geq b_{n},}
then
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≥
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{\displaystyle {1 \over n}\sum _{k=1}^{n}a_{k}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
a
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≤
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≤
⋯
≤
a
n
{\displaystyle a_{1}\leq a_{2}\leq \cdots \leq a_{n}\quad }
b
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≥
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≥
⋯
≥
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,
{\displaystyle \quad b_{1}\geq b_{2}\geq \cdots \geq b_{n},}
then
1
n
∑
k
=
1
n
a
k
b
k
≤
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1
n
∑
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=
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(
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)
.
{\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]
Consider the sum
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{\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 a j a k b j b k j , k S ≥ 0
Opening the brackets, we deduce:
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{\displaystyle 0\leq 2n\sum _{j=1}^{n}a_{j}b_{j}-2\sum _{j=1}^{n}a_{j}\,\sum _{j=1}^{n}b_{j},}
hence
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≥
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{\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 _{j=1}^{n}b_{j}\right)\!.}
An alternative proof is simply obtained with the rearrangement inequality , writing that
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{\displaystyle \sum _{i=0}^{n-1}a_{i}\sum _{j=0}^{n-1}b_{j}=\sum _{i=0}^{n-1}\sum _{j=0}^{n-1}a_{i}b_{j}=\sum _{i=0}^{n-1}\sum _{k=0}^{n-1}a_{i}b_{i+k~{\text{mod}}~n}=\sum _{k=0}^{n-1}\sum _{i=0}^{n-1}a_{i}b_{i+k~{\text{mod}}~n}\leq \sum _{k=0}^{n-1}\sum _{i=0}^{n-1}a_{i}b_{i}=n\sum _{i}a_{i}b_{i}.}
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
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{\displaystyle {\frac {1}{b-a}}\int _{a}^{b}f(x)g(x)\,dx\geq \!\left({\frac {1}{b-a}}\int _{a}^{b}f(x)\,dx\right)\!\!\left({\frac {1}{b-a}}\int _{a}^{b}g(x)\,dx\right)}
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 .
