Refined inequalities via superquadracity, overview

Since 2004 more than 80 articles were published related to superquadracity. Here we put together some of these results, in particular, those related to Jensen's inequality in one and in several variables, to Jensen-Steffensen inequalities, to Hardy's inequalities and to Hermite-Hadamard and Fejer inequalities. These inequalities are analogs of inequalities satisfied by convex functions and in those cases that the superquadratic functions are non-negative we get refinements of inequalities satisfied by convex
functions.