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

and

then

Similarly, if

and

then

[1]

Proof[edit]

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:

whence

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.

Notes[edit]

  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.