Systematic criterion for nonnegative Hermite interpolants

Develop a systematic criterion on a univariate absolutely monotonic function f that guarantees nonnegativity of its Hermite interpolant at the nodes -1, v, v, u, u used in the five-point Fibonacci-lattice certificate.

Background

The proof of the five-point result requires the univariate Hermite interpolants h1 and h2 to be nonnegative. The paper gives several sufficient inequalities, including condition (5.5) and the derivative-based condition (5.6), and verifies them for particular ranges of the worst-case integration parameter.

However, these inequalities are not presented as a general characterization. The authors explicitly acknowledge that they lack a systematic criterion that would apply broadly to the relevant class of functions.