Jensen-Type Inequalities, Montgomery Identity and Higher-Order Convexity

Motivated by some recent results known from the literature, in this paper, we establish a class of Jensen-type inequalities referring to functions of an even degree of convexity. The main idea of proving our results is a transformation of the classical Jensen functional via the Montgomery identity which is suitable to study in companion with the higher-order convexity. In such a way, we obtain superadditivity and monotonicity relations that correspond to the Jensen functional equipped with a function of an even degree of convexity. As an application, we obtain some new bounds for the differences of power means. Furthermore, we also establish some new Hölder-type inequalities. Finally, we study the Lah–Ribarič inequality in the above-described setting.