Upper bound for second derivative of cubic Hermite spline

Question

My professor mentioned the following Theorem (which we didn't prove):

Let $f \in C^4([a,b])$ and $s$ be the cubic Hermite spline interpolating $f$. Then $$ \lVert s''\rVert_\infty \leq 3 \lVert f''\rVert_\infty. $$

Do you know where this theorem is proved or do you know how to prove it?

Answer 1

Found it. A prove is described in

de Boor, Carl: A Practical Guide to Splines, revised Edition (2001), Chapter III Theorem (12), p. 34.

Although formulated for linear splines.