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.

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 was first found by Rogers (1888), and discovered independently by Hölder (1889).

The brief statement of Hölder's inequality uses some conventions.

As above, let $f$ and $g$ denote measurable real- or complex-valued functions defined on $S$ . If $|| fg || 1$ is finite, then the pointwise products of $f$ with $g$ and its complex conjugate function are $μ$ -integrable, the estimate

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