Positivity of sums and integrals for higher order nabla-convex and completely monotonic functions

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