Steffensen's Inequality

Let $f(x)$ be a Nonnegative and monotonic decreasing function in $[a,b]$ and $g(x)$ such that $0\leq g(x)\leq 1$ in $[a,b]$, 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

$$k=\int_a^b g(x)\,dx.$$

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1099, 1979.