Jensen-type inequalities in terms of Lipschitzianity

The main focus of this paper is a study of Jensen-type inequalities for the Lipschitzian functions.We establish the reverse of the Jensen inequality expressed in terms of the corresponding Lipschitz constant.In addition, we also obtain the reverse of the superadditivity relation for a convex function, expressed in the same way.As an application, we obtain reverses of power mean inequalities, the Hlder inequality, and the Hermite-Hadamard inequality, expressed in terms of the Lipschitzianity.In particular, we derive reverses of the arithmetic-geometric mean inequality in both difference and quotient forms.