Steffensen's inequality

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In mathematics, Steffensen's inequality, named after Johan Frederik Steffensen, is an integral inequality in real analysis. It states that if f : [a, b] → R is a non-negative, monotonically decreasing, integrable function and g : [a, b] → [0, 1] is another integrable function, then

\int_{b - k}^{b} f(x) \, \mathrm{d} x \leq \int_{a}^{b} f(x) g(x) \, \mathrm{d} x \leq \int_{a}^{a + k} f(x) \, \mathrm{d} x,

where

k = \int_{a}^{b} g(x) \, \mathrm{d} x.

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