Steffensen's inequality

Steffensen's inequality is an equation in mathematics named after Johan Frederik Steffensen.^[1]

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

${\displaystyle \int _{b-k}^{b}f(x)\,dx\leq \int _{a}^{b}f(x)g(x)\,dx\leq \int _{a}^{a+k}f(x)\,dx,}$

where

${\displaystyle k=\int _{a}^{b}g(x)\,dx.}$