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

Abstract

We consider the positivity of the sum ∑nρi𝟋(ξi) , where 𝟋 is a convex function of higher order, aswell as analogous results involving the integral b0 ρ(ξ)𝟋(g(ξ))dξ . We use a representation of the function 𝟋 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.