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Some very general identities of Abel and Popoviciu type for sums $\sum p_i f(x_i)$, $\sum \sum p_{; ; ; ; ; ij}; ; ; ; ; f(x_i, y_j)$ and integral $\int \int P(x, y) f(x, y) dx \, dy$ are deduced. Using obtained identities, positivity of these expressions are characterized for convex functions of higher order. An application in terms of exponential convexity is given.

Popoviciu type characterization of positivity of sums and integrals for convex functions of higher order