The Hölder inequality for sums. Let and be certain sets of complex numbers, , where is a finite or an infinite set of indices. The following inequality of Hölder is valid:
where , ; this inequality becomes an equality if and only if , and and are independent of . In the limit case, when , , Hölder's inequality has the form
If , Hölder's inequality is reversed. The converse proposition of Hölder's inequality for sums is also true (M. Riesz): If
for all such that
For sums of a more general form, Hölder's inequality takes the form
The Hölder inequality for integrals. Let be a Lebesgue-measurable set in an -dimensional Euclidean space and let the functions
belong to , condition
being satisfied. The following inequality of Hölder is then valid:
If , one obtains the Bunyakovskii inequality. Analogous remarks (concerning the sign and the limit case) as were made for the Hölder inequality
are also valid for the Hölder inequality for integrals.
In the Hölder inequality the set may be any set with an additive function (e.g. a measure) specified on some algebra of its subsets, while the functions , , are -measurable and -integrable to degree .
The generalized Hölder inequality. Let be an arbitrary set, let a (finite or infinite) functional be defined on the totality of all positive functions and let this functional satisfy the following conditions: a) ; b) for all numbers ; c) if , then the inequality is valid; and d) . If the conditions
are also met, the generalized Hölder inequality is valid for the functional:
|||O. Hölder, "Ueber einen Mittelwerthsatz" Nachr. Ges. Wiss. Göttingen (1889) pp. 38–47|
|||G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1934)|
|||E.F. Beckenbach, R. Bellman, "Inequalities" , Springer (1961)|
The Bunyakovskii inequality is better known as the Cauchy–Schwarz inequality in the English-language literature.
|[a1]||N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958)|
Hölder inequality. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=H%C3%B6lder_inequality&oldid=23328