Steffensen's inequality (original) (raw)
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Equation in mathematics
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
∫ b − k b f ( x ) d x ≤ ∫ a b f ( x ) g ( x ) d x ≤ ∫ a a + k f ( x ) d x , {\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
k = ∫ a b g ( x ) d x . {\displaystyle k=\int _{a}^{b}g(x)\,dx.}
