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Schwarz's Inequality

Let psi_1(x) and psi_2(x) be any two real integrable functions in [a,b], then Schwarz's inequality is given by

|<psi_1|psi_2>|^2<=<psi_1|psi_1><psi_2|psi_2>.
(1)

Written out explicitly

[int_a^bpsi_1(x)psi_2(x)dx]^2<=int_a^b[psi_1(x)]^2dxint_a^b[psi_2(x)]^2dx,
(2)

with equality iff psi_1(x)=alphapsi_2(x) with alpha a constant. Schwarz's inequality is sometimes also called the Cauchy-Schwarz inequality (Gradshteyn and Ryzhik 2000, p. 1099) or Buniakowsky inequality (Hardy et al. 1952, p. 16).

To derive the inequality, let psi(x) be a complex function and lambda a complex constant such that psi(x)=f(x)+lambdag(x) for some f and g. Since intpsi^_psidx>=0, where z^_ is the complex conjugate,

intpsi^_psidx=intf^_fdx+lambdaintf^_gdx+lambda^_intg^_fdx+lambdalambda^_intg^_gdx>=0,
(3)

with equality when psi(x)=0. Writing this in compact notation,

<f^_,f>+lambda<f^_,g>+lambda^_<g^_,f>+lambdalambda^_<g^_,g>>=0.
(4)

Now define

lambda=-(<g^_,f>)/(<g^_,g>)
(5)
lambda^_=-(<g,f^_>)/(<g^_,g>).
(6)

Multiply (4) by <g^_,g> and then plug in (5) and (6) to obtain

<f^_,f><g^_,g>-<f^_,g><g^_,f> -<g^_,f><g,f^_>+<g^_,f><g,f^_>>=0,
(7)

which simplifies to

<g^_,f><f^_,g><=<f^_,f><g^_,g>
(8)

so

|<f,g>|^2<=<f,f><g,g>.
(9)

Bessel's inequality follows from Schwarz's inequality.

See also

Bessel's Inequality, Cauchy's Inequality, Hölder's Inequalities

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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.Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 527-529, 1985.Buniakowsky, V. "Sur quelques inégalités concernant les intégrales ordinaires et les intégrales aux différences finies." Mémoires de l'Acad. de St. Pétersbourg (VII) 1, No. 9, p. 4, 1959.Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1099, 2000.Hardy, G. H.; Littlewood, J. E.; and Pólya, G. "Further Remarks on Method: The Inequality of Schwarz." §6.5 in Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 132-134, 1952.Schwarz, H. A. "Über ein die Flächen kleinsten Flächeninhalts betreffendes Problem der Variationsrechnung." Acta Soc. Scient. Fen. 15, 315-362, 1885. Reprinted in Gesammelte Mathematische Abhandlungen, Vol. 1. New York: Chelsea, pp. 224-269, 1972.

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Schwarz's Inequality

Cite this as:

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

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