Positivity of sums and integrals for n-convex functions via the Fink identity and new Green functions

We consider positivity of sum (Formula Presented)) involving convex functionsof higher order. Analogous for integral (Formula Presented) is also given. Represen-tation of a function f via the Fink identity and the Green function leads us toidentities for which we obtain conditions for positivity of the mentioned sum andintegral. We obtain bounds for integral remainders which occur in those identitiesas well as corresponding mean value theorems © 2021, Studia Universitatis Babes-Bolyai Mathematica. All Rights Reserved.