Steffensen's inequality

Steffensen's inequality bounds the integral of a monotone function weighted by a function between zero and one, using the weight's total integral. It compares that weighted integral with integrals over intervals of matching length.

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Integral rearrangement inequality: An inequality comparing integrals of products based on how functions’ values are arranged. Steffensen’s bounds arise from arranging the weight’s mass at either endpoint.
Lebesgue integral: An integral defined by measuring function values across measurable sets. The inequality extends naturally from ordinary integrals to measurable functions.
Numerical integration: The approximation of definite integrals using finite computations. Monotone integrands allow weighted sampling schemes to be bounded by interval integrals.
Johan Frederik Steffensen: A Danish mathematician known for work in interpolation, inequalities, and numerical analysis. The inequality bears his name and reflects his work on integral bounds.
Chebyshev's integral inequality: An inequality bounding the mean of a product of similarly or oppositely ordered functions. It also exploits monotonicity, but bounds a product integral through correlation rather than endpoint intervals.
Riemann integral: An integral defined as a limit of sums over partitions of an interval. Classical statements often use interval integrals before broader measure-theoretic formulations.
Integral inequalities: Inequalities that constrain integrals using properties of their integrands or domains. Steffensen’s result is a reusable bound within this broad family.
Jensen's inequality: An inequality relating a convex function of an average to the average of the function. It is a foundational classical inequality often used alongside integral bounds.
Hardy–Littlewood rearrangement inequality: An inequality relating integrals of products to integrals of functions rearranged in monotone order. It gives a broader rearrangement framework for the extremal placement of weight.
Almost everywhere: A property holds almost everywhere when its exceptions form a set of measure zero. Measurable versions need monotonicity or bounds only up to null sets.