6
$\begingroup$

I wonder if there is a name or reference for the following fact. It is not the proof I am looking for.

Let $s_1, s_2, ...,s_n$ be non-negative real numbers ordered in a non-increasing way. Let $b_1,b_2,...,b_n$ be non-negative real numbers ordered in a non-decreasing way (so opposite of the $s_i$).

Then the average value $$\frac{s_1+s_2+\cdots +s_n}{n}$$ is at least as large as the weighted average $$\frac{b_1 s_1+\cdots+b_n s_n}{b_1+\cdots+b_n}.$$

Thanks!

$\endgroup$
3
  • 12
    $\begingroup$ Chebyshev's sum inequality $\endgroup$ Mar 8, 2018 at 19:57
  • $\begingroup$ Please use TeX on this site. $\endgroup$
    – GH from MO
    Mar 8, 2018 at 20:12
  • 2
    $\begingroup$ I thought it looked like the "Rearrangement Inequality" (en.wikipedia.org/wiki/Rearrangement_inequality); it's not quite, but the inequality here can be recovered from it by averaging over all $n!$ rearrangements. (The Wikipedia page notes that Chebyshev's sum inequality is a consequence of Rearrangement.) $\endgroup$ Mar 9, 2018 at 4:24

2 Answers 2

19
$\begingroup$

This is known as Chebyshev's Sum Inequality. (I've only ever seen it used in the context of competition math, but Wikipedia gives a reference to Hardy-Littlewood-Polya.)

$\endgroup$
1
  • 2
    $\begingroup$ I've seen it used at least once by K. Matomäki (in "On signs of Fourier coefficients of cusp forms", Math. Proc. Cambridge Philos. Soc. 152 (2012), no. 2, 207–222, if I remember correctly). $\endgroup$ Mar 9, 2018 at 9:07
4
$\begingroup$

It is called often called "Chebyshev's other inequality". And this makes more sense than the name "Chebyshev's sum inequality" proposed by Wikipedia, since even Wikepedia knows that there is a version for integrals.

$\endgroup$
3
  • 7
    $\begingroup$ Integrals aren’t sums? $\endgroup$ Mar 9, 2018 at 0:06
  • $\begingroup$ So one of my physics teachers told us. But I could understand it only after a transposition. $\endgroup$ Mar 9, 2018 at 0:14
  • $\begingroup$ @LutzMattner : ... so integrals are permutations of sums? :-) $\endgroup$ Mar 9, 2018 at 2:07

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.