Direct conditions guaranteeing a valid two-variable interpolant

Determine conditions on a two-variable potential γ that guarantee its Hermite interpolant at the five-point Fibonacci-lattice interpolation nodes lies below γ, preferably through direct assumptions on γ and its derivatives rather than separate conditions on tensor factors.

Background

The paper’s rigorous certificate relies on a tensor-product representation γ(s,t)=f1(s)f2(t), together with positivity conditions on the individual factors and their Hermite interpolants. For general two-variable potentials, the authors instead consider direct Hermite interpolation at the relevant nodes, but they do not know a condition ensuring that the interpolant remains below the potential. They also show that nonnegative mixed derivatives alone are insufficient, motivating the search for stronger structural hypotheses such as logarithmic convexity or related properties.