The finite-difference characterization of summability for completely convex functions

Let $\mathbb{R}_+=(0,\infty)$, let $\mathcal{K}^{\infty}$ denote the class of functions that are completely convex, and let $\mathcal{D}^{\infty}_{\mathbb{N}}$ denote the class of functions for which some finite forward difference sequence converges. For a function $g\colon\mathbb{R}_+\to\mathbb{R}$ that is not eventually identically zero, consider condition

. **Finite-difference characterization.** If $g$ lies in $\mathcal{K}^{\infty}$ and is not eventually identically zero, then $g$ lies in $\mathcal{D}^{\infty}_{\mathbb{N}}$ if and only if condition

holds. The condition is motivated by examples such as $g(x)=2^x$ and $g(x)=\Gamma(x)$, for which it fails when $g\in\mathcal{K}^{\infty}\setminus\mathcal{D}^{\infty}_{\mathbb{N}}$; the source does not resolve the conjecture.