Direct conditions for bivariate Hermite interpolants

Determine conditions on a bivariate potential gamma that guarantee its Hermite interpolant at the five-point Fibonacci interpolation nodes lies below gamma, preferably using direct conditions on the potential and its derivatives rather than separate conditions on tensor factors.

Background

For tensor-product potentials gamma(s,t)=f1(s)f2(t), the paper controls the Hermite interpolants of the individual factors and combines them with auxiliary functions. This approach does not directly handle general bivariate potentials in a systematic way.

The authors explicitly state that even nonnegative mixed derivatives do not suffice to ensure that the direct bivariate Hermite interpolant remains below the potential. They leave unresolved the identification of more effective hypotheses, possibly involving logarithmic convexity or related properties.