New Steffensen-type inequalities are obtained by combining generalized Taylor expansions, Rabier and Pečarić extensions of Steffensen’s inequality Faà di Bruno’s formula for higher order derivatives the composition.

Steffensen’s inequality deals with the comparison between integrals over a whole interval [a, b] and integrals over a subset of [a, b]. In this paper we prove an inequality which is similar to Steffensen’s inequality. The most general form of this inequality deals with integrals over a measure space. We also consider the discrete case.