Let
and
be any two realintegrable
functions in ,
then Schwarz's inequality is given by
(1)
Written out explicitly
(2)
with equality iff with 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).
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.