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 $(S, Σ, μ)$ 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 $f \in L p (μ)$ and $g \in L q (μ)$ , then Hölder's inequality becomes an equality if and only if $| f | p$ and $| g | q$ are linearly dependent in $L 1 (μ)$ , meaning that there exist real numbers $α, β \geq 0$ , not both of them zero, such that $α | f | p = β | g | q$ $μ$ -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 $L p (μ)$ and $g$ is in $L q (μ)$ , then the pointwise product $fg$ is in $L 1 (μ)$ .

Hölder's inequality is used to prove the Minkowski inequality, which is the triangle inequality in the space $L p (μ)$ , and also to establish that $L q (μ)$ is the dual space of $L p (μ)$ 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.

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