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

Abstract

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