A remark on the connection between the second divided difference and the second derivative

In the recent paper of S.I. Novikov and V.T. Shevaldin, the problem of the relationship between the second divided difference and the second derivative has been considered. The problem is to find the smallest value (in the uniform norm) of the second derivative among the functions interpolating a sequence of values with bounded second divided differences on arbitrary grids. In their paper, two-sided estimates for the required quantity have been found. We note that a more exact upper bound is known; it is attainable, for example, on a uniform grid. This bound can be easily obtained using Subbotin's interpolation splines.