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

$\underset{b-k}{\overset{b}{\int }}f(x)dx\le \underset{a}{\overset{b}{\int }}f(x)g(x)dx\le \underset{a}{\overset{a+k}{\int }}f(x)dx,$

where

$k=\underset{a}{\overset{b}{\int }}g(x)dx.$

External links

• Weisstein, Eric W.. "Steffensen's Inequality". http://mathworld.wolfram.com/SteffensensInequality.html.