In this paper generalized Steffensen type inequalities related to the class of functions that are “convex at pointc” are derived and as a consequence inequalities involving the class of convex functions are obtained. Moreover, linear functionals from the difference of the right- and left-hand side of the obtained generalized inequalities are constructed and new families of exponentially convex functions related to constructed functionals are derived.

In this paper, we obtain some new weighted Hermite–Hadamard-type inequalities for $(n+2)-$convex functions by utilizing generalizations of Steffensen’s inequality via Taylor’s formula.

In this paper, we extend some Steffensen-type inequalities to time scales by using the diamond-$\alpha$-dynamic integral. Further, we prove some new Steffensen-type inequalities for convex functions utilizing positive $\sigma$-finite measures in time scale calculus. Moreover, as a special case, we obtain these inequalities for the delta and the nabla integral. By using the relation between calculus on time scales $\mathbb{T}$ and differential calculus on $\mathbb{R}$, we obtain already-known Steffensen-type inequalities.