%0 Journal Article %T Positivity of Integrals for Higher Order $\nabla-$Convex and Completely Monotonic Functions %J Sahand Communications in Mathematical Analysis %I University of Maragheh %Z 2322-5807 %A Mehmood, Faraz %A Khan, Asif Raza %A Adnan, Muhammad %D 2022 %\ 02/01/2022 %V 19 %N 1 %P 119-137 %! Positivity of Integrals for Higher Order $\nabla-$Convex and Completely Monotonic Functions %K Convex functions %K $nabla-$convex functions %K completely monotonic functions %R 10.22130/scma.2021.527460.926 %X 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. %U https://scma.maragheh.ac.ir/article_249585_172a7aeb2e4fe6ea55ee687557aa2805.pdf