Steffensen's inequality

In mathematics, Steffensen's inequality, named after Johan Frederik Steffensen, is an integral inequality in real analysis. It states that 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

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

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