A note on some bounds between cubic spline interpolants depending on the boundary conditions: Application to a monotonicity property

In the context of cubic splines, the authors have contributed to a recent paper dealing with the computation of nonlinear derivatives at the interior nodes so that monotonicity is enforced while keeping the order of approximation of the spline as high as possible. During the review process of that paper, one of the reviewers raised the question of whether a cubic spline interpolating monotone data could be forced to preserve monotonicity by imposing suitable values of the first derivative at the endpoints. Albeit a negative answer appears to be intuitive, we have found no results regarding this fact. In this short work we prove that the answer to that question is actually negative.