structural:expansion, structural:reformatting
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 spaces.

Hölder's inequality — Let (S, Σ, μ) be a measure space and let p, q ∈ with 1/ p + 1/ q = 1. Then for all measurable real - or complex -valued functions f and g on S,

‖ f g ‖ 1 ≤ ‖ f ‖ p ‖ g ‖ q. {\displaystyle \|fg\|_{1}\leq \|f\|_{p}\|g\|_{q}.}

If, in addition, p, q ∈ (1, ∞) and f ∈ L (μ) and g ∈ L (μ), then Hölder's inequality becomes an equality if and only if | f | and | g | are linearly dependent in L (μ), meaning that there exist real numbers α, β ≥ 0, not both of them zero, such that α | f | = β | g | 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 ‖ is infinite, the right-hand side also being infinite in that case. Conversely, if f is in L (μ) and g is in L (μ), then the pointwise product fg is in L (μ).

Hölder's inequality is used to prove the Minkowski inequality, which is the triangle inequality in the space L (μ), and also to establish that L (μ) is the dual space of L (μ) for p ∈ [ 1, ∞).

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.