In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality relating L^p spaces: let S be a measure space, let 1 ≤ p, q ≤ ∞ with 1/p + 1/q = 1, let f be in L^p(S) and g be in L^q(S). Then fg is in L¹(S) and

$\backslash |fg\backslash |\_1\; \backslash le\; \backslash |f\backslash |\_p\; \backslash |g\backslash |\_q.$

The numbers p and q above are said to be Hölder conjugates of each other.

Hölder's inequality is used to prove the generalization of the triangle inequality in the space L^p, the Minkowski inequality, and also to establish that L^p is dual to L^q.

Notable special cases

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• For p = q = 2 Holder's inequality results in the Cauchy-Schwarz inequality.

• In the case of the Euclidean space, when set S is {1,...,n} with the counting measure, one has that for all x, y in Rⁿ (Cⁿ)

$\backslash sum\_\{k=1\}^n\; |x\_k\; y\_k|\; \backslash leq\; \backslash left(\; \backslash sum\_\{k=1\}^n\; |x\_k|^p\; \backslash right)^\{1/p\}\; \backslash left(\; \backslash sum\_\{k=1\}^n\; |y\_k|^q\; \backslash right)^\{1/q\}$

• If S=N with the counting measure, then we get Holder's inequality for sequences from lp spaces

$\backslash sum\backslash limits\_\{n=1\}^\{\backslash infty\}\; |x\_n\; \backslash cdot\; y\_n|\; \backslash le\; \backslash left(\; \backslash sum\backslash limits\_\{n=1\}^\{\backslash infty\}\; |x\_n|^p\; \backslash right)^\{1/p\}\; \backslash cdot\; \backslash left(\; \backslash sum\backslash limits\_\{n=1\}^\{\backslash infty\}\; |y\_n|^q\; \backslash right)^\{1/q\},\backslash ;\; \backslash forall\; x\; \backslash in\; l^p,\; y\backslash in\; l^q$.

• For the space of integrable complex-valued functions, one has

$\backslash left|\backslash int\; f(x)g(x)\backslash ,dx\backslash right|\backslash leq\backslash left(\backslash int\; \backslash left|f(x)\backslash right|^p\backslash ,dx\; \backslash right)^\{1/p\}\backslash cdot\; \backslash left(\backslash int\backslash left|g(x)\backslash right|^q\backslash ,dx\backslash right)^\{1/q\}.$

• For the probability space $(\backslash Omega,\backslash mathcal\{F\},\backslash mathbb\{P\})$, $L^p(\backslash Omega,\backslash mathcal\{F\},\backslash mathbb\{P\})$ denotes the space of the random variables with finite p-moment, , where the symbol $\backslash mathbb\{E\}$ denotes the expected value. Holder's inequality becomes

$\backslash mathbb\{E\}|XY|\; \backslash le\; \backslash left(\backslash mathbb\{E\}|X|^p\backslash right)^\{1/p\}\; \backslash cdot\; \backslash left(\; \backslash mathbb\{E\}|Y|^q\; \backslash right)^\{1/q\},\backslash ;\; \backslash forall\; X\; \backslash in\; L^p,\; Y\; \backslash in\; L^q$.

Generalization

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The following generalization can be proven by induction.

Assume $p\_k\backslash geq\; 1,\; k=1,\backslash ldots\; n$ are such that

$\backslash sum\_\{k=1\}^n\; \backslash frac\{1\}\{p\_k\}=1$

Assume that $u\_k\backslash in\; L^\{p\_k\}(S)$. Then $\backslash prod\_\{k=1\}^n\; u\_k\; \backslash in\; L^1(S)$ and

$\backslash left\backslash |\backslash prod\_\{k=1\}^n\; u\_k\backslash right\backslash |\_\{\backslash displaystyle\; L^1(S)\}\backslash leq\; \backslash prod\_\{k=1\}^n\; \backslash |u\_k\backslash |\_\{\backslash displaystyle\; L^\{p\_k\}(S)\}$

Inequalities | Functional analysis

Hölder-Ungleichung | Inégalité de Hölder | Nierówność Höldera | Неравенство Гёльдера | Hölder-egyenlőtlenség | Hölder Eşitsizliği | 赫尔德不等式