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Chebyshev Sum Inequality

If

a_1>=a_2>=...>=a_n
(1)
b_1>=b_2>=...>=b_n,
(2)

then

nsum_(k=1)^na_kb_k>=(sum_(k=1)^na_k)(sum_(k=1)^nb_k).
(3)

This is true for any distribution.

See also

Cauchy's Inequality, Chebyshev Inequality, Hölder's Inequalities

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References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1092, 2000.Hardy, G. H.; Littlewood, J. E.; and Pólya, G. Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 43-44, 1988.

Referenced on Wolfram|Alpha

Chebyshev Sum Inequality

Cite this as:

Weisstein, Eric W. "Chebyshev Sum Inequality." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ChebyshevSumInequality.html

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