Steffensen's inequality

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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

\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

^ Rabier, Patrick J. (2012). "Steffensen's inequality and $L^{1}$ – $L$ ^∞ estimates of weighted integrals". Proceedings of the American Mathematical Society. 140 (2): 665–675. doi:10.1090/S0002-9939-2011-10939-0. ISSN 0002-9939.

External links

Weisstein, Eric W. "Steffensen's Inequality". MathWorld.

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