Positivity of sums and integrals for higher order nabla-convex and completely monotonic functions

TL;DR

This paper generalizes the concepts of nabla-convexity and complete monotonicity for functions of two variables, deriving identities and conditions for positivity of sums and integrals, with applications to means and convexity.

Contribution

It introduces extended definitions for higher order nabla-convex and completely monotonic functions, along with identities and positivity characterizations for sums and integrals.

Findings

Derived identities of Popoviciu type for sums and integrals.

Characterized positivity conditions for higher order nabla-convex and monotonic functions.

Presented applications to generalized means and exponential convexity.

Abstract

We extend the definitions of $\nabla-$convex and completely monotonic functions for two variables. Some general identities of Popoviciu type for sum $\sum \sum p_{ij} f(y_i, z_j)$ and integrals $\int P(y)f(y) dy$, $\int \int P(y,z) f(y,z) dy \, dz$ 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.