  We extend the definitions of $\nabla-$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 $\nabla-$convex and
completely monotonic functions. Some applications in terms of generalized
Cauchy means and exponential convexity are given.
