Schwarz's Inequality
Let 
and 
be any
two real integrable
functions in ![[a,b]](/images/equations/SchwarzsInequality/Inline3.gif)
, then Schwarz's inequality is given
by
 |
(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,](/images/equations/SchwarzsInequality/NumberedEquation2.gif) |
(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).
To derive the inequality, let 
be a complex
function and 
a complex
constant such that 
for some 
and 
. Since 
,
where 
is the complex
conjugate,
 |
(3)
|
with equality when 
. Writing this in compact notation,
 |
(4)
|
Now define
Multiply (4) by 
and
then plug in (5) and (6) to obtain
 |
(7)
|
which simplifies to
 |
(8)
|
so
 |
(9)
|
Bessel's inequality follows from Schwarz's
inequality.
SEE ALSO: Bessel's Inequality,
Cauchy's Inequality,
Hölder's
Inequalities
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
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