Hölder's inequality

In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of $L p$ spaces.

Hölder's inequality—Let $(S,\Sigma ,\mu )$ be a measure space and let $p,q\in [1,\infty ]$ with $1/ p + 1/ q = 1$ . Then for all measurable real- or complex-valued functions $f$ and $g$ on $S$ ,

$\|fg\|_{1}\leq \|f\|_{p}\|g\|_{q}.$

If, in addition, $p,q\in (1,\infty )$ and $f\in L^{p}(\mu )$ and $g\in L^{q}(\mu )$ , then Hölder's inequality becomes an equality if and only if $|f|^{p}$ and $|g|^{q}$ are linearly dependent in $L^{1}(\mu )$ , meaning that there exist real numbers $\alpha ,\beta \geq 0$ , not both of them zero, such that $\alpha |f|^{p}=\beta |g|^{q}$ $\mu$ -almost everywhere.

The numbers $p$ and $q$ above are said to be Hölder conjugates of each other. The special case $p=q=2$ gives a form of the Cauchy–Schwarz inequality. Hölder's inequality holds even if $\|fg\|_{1}$ is infinite, the right-hand side also being infinite in that case. Conversely, if $f$ is in $L^{p}(\mu )$ and $g$ is in $L^{q}(\mu )$ , then the pointwise product $fg$ is in $L^{1}(\mu )$ .

Hölder's inequality is used to prove the Minkowski inequality, which is the triangle inequality in the space $L^{p}(\mu )$ , and also to establish that $L^{q}(\mu )$ is the dual space of $L^{p}(\mu )$ for $p\in [1,\infty )$ .

Hölder's inequality (in a slightly different form) was first found by Leonard James Rogers (1888). Inspired by Rogers' work, Hölder (1889) gave another proof as part of a work developing the concept of convex and concave functions and introducing Jensen's inequality, which was in turn named for work of Johan Jensen building on Hölder's work.