## Steffensen's inequality proof

Posted December 10th, 2011 by MSA

Johan Frederik Steffensen (1873–1961)

ƒ : [a, b] → R is a non-negative, monotonically decreasing, integrable function and

g:[a, b] →[0, 1] is another integrable function, then:

where

First we see that and . We have and

so

We have also

take and and so

That proof the inequality.