Generalizations of Jensen-Steffensen and related integral inequalities for superquadratic functions

We present integral versions of some recently proved results which improve the Jensen-Steffensen and related inequalities for superquadratic functions. For superquadratic functions which are not convex we get inequalities analogous to the integral Jensen-Steffensen inequality for convex functions. Therefore, we get refinements of all the results which use only the convexity of these functions. One of the inequalities that we obtain for a superquadratic function φ is, where, and, which under suitable conditions like those satisfied by functions of power equal or more than 2, is a refinement of the Jensen-Steffensen-Boas inequality. We also prove related results of Mercer's type.