: Mathématiques: Steffensen's inequality

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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

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

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

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