Steffensen's inequality

{{Short description|Equation in mathematics}}

{{single source|date=August 2007}}

Steffensen's inequality is an equation in mathematics named after Johan Frederik Steffensen.{{cite journal|last1=Rabier|first1=Patrick J.|title=Steffensen's inequality and $L^\{1\}$ – $L$^∞ estimates of weighted integrals|journal=Proceedings of the American Mathematical Society|volume=140|issue=2|year=2012|pages=665–675|issn=0002-9939|doi=10.1090/S0002-9939-2011-10939-0|doi-access=free}}

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

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

:where

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