Positivity of Integrals for Higher Order $nabla-$Convex and Completely Monotonic Functions

We extend the definitions of $nabla-$convex and completely monotonic functions for two variables. Some general identities of Popoviciu type integrals $int P(y)f(y) dy$ and $int int P(y,z) f(y,z) dy   dz$ are deduced. Using obtained identities, positivity of these expressions are characterized for  higher order $nabla-$convex and completely monotonic functions. Some applications in terms of generalized Cauchy means and exponential convexity are given.