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.

Keywords:

Convex functions , nabla-convex functions , completely monotonic functions