Document Type : Research Paper


1 Department of Mathematics, Dawood University of Engineering and Technology, New M. A. Jinnah Road, Karachi-74800, Pakistan.

2 Department of Mathematics, University of Karachi, University Road, Karachi-75270 Pakistan.


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.


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