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Chebyshev's sum inequality

Mathematical inequality From Wikipedia, the free encyclopedia

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

and

then

In words, if we are given two sequences that are both non-increasing or non-decreasing, then the product of their averages is less than the average of their (termwise) product.

Similarly, if

and

then

[1]
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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 j, k. Hence S  0.

Opening the brackets, we deduce:

hence

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

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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.

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See also

Notes

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