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 ?
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.
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.
To prove this inequality, we will use the improvement of Holder's inequality from relation (10).
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.