Abstract The main objective of this research is to characterize all the real polynomial functions of degree less than the fourth which are Jensen m -convex on the set of non-negative real numbers. In the first section, it is established for that class of functions what conditions must satisfy a particular polynomial in order to be starshaped on the same set. Finally, both kinds of results are combined in order to find examples of either Jensen m -convex functions which are not starshaped or viceversa.