Upper bounds for the second order derivatives convex spline interpolation

For data sets in convex position we derive upper bounds for the second order derivatives which are fair in convex interpolation. We show that there are spline functions such that convex interpolation is always successful even when, in addition, the derived bounds are active. Upper bounds on the second order derivatives are of interest in order to avoid large curvatures which may considerably disturb the visual pleasantness of the interpolants.