Refining the Superadditivity Inequality for Weighted Jensen and Mercer Functionals

The Jensen and Mercer functionals viewed as functions of their weighted vectors are superadditive as showed by S. S. Dragomir, J. E. Pečarić and L. E. Persson [Properties of some functionals related to Jensen's inequality, Acta Math. Hung. 70/1-2 (1996) 129–143] and by M. Krnić, N. Lovričević and J. Pečarić [On some properties of Jensen-Mercer's functional, J. Math. Inequalities 6/1 (2012) 125–139]. In the present paper we derive refinements of the corresponding superadditivity DPP/KLP inequalities. We establish Jensen and Mercer inequalities of second type. We introduce a preorder on the set of weighted vectors and show the monotonicity with respect to this preorder of a functional related to Jensen/Mercer inequality.