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

We consider the positivity of the sum Σ i=1 ; n ρ i F(ξ i ), where F is a convex function of higher order, as well as analogous results involving the integral ∫ a0 b0 ρ(ξ)F(g(ξ))dξ . We use a representation of the function F via the Fink identity and the Green function that leads us to identities from which we obtain conditions for positivity of the above-mentioned sum and integral. We also obtain bounds for the integral remainders which occur in these identities, as well as corresponding mean value results. © Tübitak.