On the Bellman generalization of Steffensen's inequality

Abstract

Steffensen [ 1 ] proved the following result: Assume that two integrable functions f(t) and g(t) are defined on the interval [a, b], that f(t) never increases, and that 0 <g(t) < 1 in [Q, 61. Then Using the substitution g(t) = LG(t)/(f: G(t) dt, we have the following modification of the Steffensen result: Assume that two integrable functions f(t) and G(t) are defined on the interval [a, b], thatf(t) never increases, and that 0 < dG(t) <j” G(t) dt (VlE [a,bl), (2) n where k is a positive number. Then This result is an extension of Theorem 2 from [2]. If A= (b-(q-1