Steffensen's inequality

Steffensen's inequality is an equation in mathematics named after Johan Frederik Steffensen.

It is an integral inequality in real analysis, stating:

If ƒ : [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) \, dx \leq \int_{a}^{b} f(x) g(x) \, dx \leq \int_{a}^{a + k} f(x) \, dx,

where

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

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