# Hölder's Inequality

(redirected from*Holder'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

│*a*_{1}*b*_{1} + … + *a _{n}b_{n}* │

≤(│*a*_{1} │ ^{P} + … + │ *a _{n}* │

*)*

^{p}^{1/p}(│

*b*

_{1}│

^{q}+ … + │

*b*│

_{n}^{q}+ … + │

*b*│

_{n}^{q})

^{1/q}

and for integrals,

│∫*g(x)h(x) dx* │ ≤[ ∫ │*g(x)*│ ^{p}*dx*]^{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.