Hölder's Inequality

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Hölder's inequality

[′hel·dərz ‚in·i′kwäl·əd·ē]
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 ?
Using Holder's inequality, to get the desired estimate is enough to estimate the integral
Using the following Holder's inequality for the double integral:
of Pennsylvania) starts with Cauchy and progresses to the AM-GM inequality, Lagrange's identity and Minkowski's conjecture, geometry and sums of squares, the consequences of order, convexity, integrals, power means, Holder's inequality, Hilbert's inequality, symmetric sums, majorization and Schur convexity, cancellation and aggregation.
We prove our main results by using the time scales chain rule, the time scales integration by parts formula, the time scales Taylor formula, and classical as well as time scales versions of Holder's inequality.
By the Holder's inequality and Sobolev embedding inequality, we have
a) It is easy to see that [greater than or equal to] > 1 and using inequality (10) we have a refinement of Holder's inequality.
Now, applying Holder's inequality with indices k and k', where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Whenever k > 1, we can apply Holder's inequality with indices k and k', where 1/k + 1/k' = 1.
Keywords and Phrases: Opial type inequality, Finite difference inequalities, Forward differences, Holder's inequality, Nondecreasing functions of several variables.