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Cauchy's Inequality

A special case of Hölder's sum inequality with p=q=2,

(sum_(k=1)^na_kb_k)^2<=(sum_(k=1)^na_k^2)(sum_(k=1)^nb_k^2),
(1)

where equality holds for a_k=cb_k. The inequality is sometimes also called Lagrange's inequality (Mitrinović 1970, p. 42), and can be written in vector form as

||a·b||<=||a||||b||.
(2)

In two-dimensions, it becomes

(a^2+b^2)(c^2+d^2)>=(ac+bd)^2.
(3)

It can be proven by writing

sum_(i=1)^n(a_ix+b_i)^2=sum_(i=1)^na_i^2(x+(b_i)/(a_i))^2=0.
(4)

If b_i/a_i is a constant c, then x=-c. If it is not a constant, then all terms cannot simultaneously vanish for real x, so the solution is complex and can be found using the quadratic equation

x=(-2suma_ib_i+/-sqrt(4(suma_ib_i)^2-4suma_i^2sumb_i^2))/(2suma_i^2).
(5)

In order for this to be complex, it must be true that

(sum_(i)a_ib_i)^2<=(sum_(i)a_i^2)(sum_(i)b_i^2),
(6)

with equality when b_i/a_i is a constant. The vector derivation is much simpler,

(a·b)^2=a^2b^2cos^2theta<=a^2b^2,
(7)

where

a^2=a·a=sum_(i)a_i^2,
(8)

and similarly for b.

See also

Chebyshev Inequality, Chebyshev Sum Inequality, Hölder's Inequalities, Schwarz's Inequality

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 11, 1972.Apostol, T. M. Calculus, 2nd ed., Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. Waltham, MA: Blaisdell, pp. 42-43, 1967.Cauchy, A. L. Cours d'analyse de l'École Royale Polytechnique, 1ère partie: Analyse algébrique. Paris: p. 373, 1821. Reprinted in Œuvres complètes, 2e série, Vol. 3.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. "Cauchy's Inequality." §2.4 in Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 16-18, 1952.Jeffreys, H. and Jeffreys, B. S. "Cauchy's Inequality." §1.16 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 54, 1988.Krantz, S. G. Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 12, 1999.Mitrinović, D. S. "Cauchy's and Related Inequalities." §2.6 in Analytic Inequalities. New York: Springer-Verlag, pp. 41-48, 1970.

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Cauchy's Inequality

Cite this as:

Weisstein, Eric W. "Cauchy's Inequality." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CauchysInequality.html

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