Hölder's Inequality

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Hölder's inequality

[′hel·dərz ‚in·i′kwäl·əd·ē]
(mathematics)
Generalization of the Schwarz inequality: for real functions |∫ƒ(x) g (x) dx | ≤ (∫|ƒ(x)| p dx)1/ p · (∫| g (x)| q dx)1/ q where 1/ p + 1/ q = 1.

Hölder’s Inequality

 

For finite sums, it is

a1b1 + … + anbn

≤(│a1P + … + │ anp)1/p (│ b1q + … + │ bnq + … + │ bnq) 1/q

and for integrals,

│∫g(x)h(x) dx │ ≤[ ∫ │g(x)pdx]1/p[∫ │h(x)q dx]1/q

where p > 1 and 1/p + 1/q = 1. Hölder’s inequality was established by the German mathematician O. L. Hölder in 1889 and is one of the most commonly used in mathematical analysis. For p = q = 2 it is transformed for finite sums into Cauchy’s inequality and for integrals, into Buniakovskii’s inequality.

References in periodicals archive ?
([13], [17]) For u [member of] [L.sup.p(x)([OMEGA])] and v [member of] [L.sup.p'(x)]([OMEGA]), we have the Holder's inequality
For 1/r + 1/q = 1/2, the Holder's inequality and Gagliardo-Nirenberg inequality yield
As p [greater than or equal to] 1, by Holder's inequality we have that
Applying successively Holder's inequality and Fubini's Theorem, we obtain that
Moreover, by using Holder's inequality and Lemma 3.2, we obtain
Note that the direction of Holder's inequality is reverse of the usual one for p < 1.
Using Lemma 7, Holder's inequality, and the fact that [[absolute value of (f')].sup.q] is an m-convex function,
Using the following Holder's inequality for the double integral:
Define z = [u.sup.p/2], by Holder's inequality; (15) can be written as
Actually, suppose [q.sub.i] > 0 (i = 1, 2, 3) such that [1/[q.sub.1]] + [1/[q.sub.2]] + (1/[q.sub.3]) = 1, it follows from Holder's inequality that