Hölder Integral Inequality

If

$${1\over p}+{1\over q}=1$$

with $p$, $q>1$, then

$$\int_a^b \vert f(x)g(x)\vert\,dx \leq \left[{\int_a^b \vert ... ...t]^{1/p} \left[{\int_a^b \vert g(x)\vert^q \,dx}\right]^{1/q},$$

with equality when

$$\vert g(x)\vert=c\vert f(x)\vert^{p-1}.$$

If $p=q=2$, this inequality becomes Schwarz's Inequality.

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 11, 1972.

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1099, 1993.

Hölder, O. ``Über einen Mittelwertsatz.'' Göttingen Nachr., 44, 1889.

Riesz, F. ``Untersuchungen über Systeme integrierbarer Funktionen.'' Math. Ann. 69, 456, 1910.

Riesz, F. ``Su alcune disuguaglianze.'' Boll. Un. Mat. It. 7, 77-79, 1928.

Sansone, G. Orthogonal Functions, rev. English ed. New York: Dover, pp. 32-33, 1991.