Hölder's Inequality


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 │

≤(│a1 │ P + … + │ an │ p)1/p (│ b1 │ q + … + │ bn │ q + … + │ bn │ q) 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.