In this paper some Bellman-Steffensen type inequalities are generalized for positive measures. Using sublinearity of a class of convex functions and Jensen's inequality, nonnormalized versions of Steffensen's inequality are obtained. Further, linear functionals, from obtained Bellman-Steffensen type inequalities, are produced and their action on families of exponentially convex functions is studied.
Keywords: Bellman–Steffensen type inequality; Exponential convexity; Measure theory; Steffensen’s inequality.