Positivity of Sums and Integrals for n-Convex Functions via Abel-Gontscharoff's Interpolating Polynomial and Green Functions

We consider positivity of sum Sigma(n)(i=1) p(i)f(x(i)) involving convex functions of higher order. Analogous for integral integral(b)(a) p(x)f(g(x))dx is also given. Representation of a function f via the Abel-Gontscharoff's Interpolating Polynomial and Green functions leads us to identities for which we obtain conditions for positivity of the mentioned sum and integral. We obtain bound for integral remainders which occur in those identities as well as corresponding mean value theorem.