TY - JOUR ID - 249585 TI - Positivity of Integrals for Higher Order $\nabla-$Convex and Completely Monotonic Functions JO - Sahand Communications in Mathematical Analysis JA - SCMA LA - en SN - 2322-5807 AU - Mehmood, Faraz AU - Khan, Asif Raza AU - Adnan, Muhammad AD - Department of Mathematics, Dawood University of Engineering and Technology, New M. A. Jinnah Road, Karachi-74800, Pakistan. AD - Department of Mathematics, University of Karachi, University Road, Karachi-75270 Pakistan. Y1 - 2022 PY - 2022 VL - 19 IS - 1 SP - 119 EP - 137 KW - Convex functions KW - $nabla-$convex functions KW - completely monotonic functions DO - 10.22130/scma.2021.527460.926 N2 - 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. UR - https://scma.maragheh.ac.ir/article_249585.html L1 - https://scma.maragheh.ac.ir/article_249585_172a7aeb2e4fe6ea55ee687557aa2805.pdf ER -