Let $x_1 < \cdots < x_k \in \mathbb{R}$ be a equidistant subdivision of the interval $I = [x_1,x_k]$ and let $\{(x_1, y_1), \ldots, (x_k, y_k)\} \subseteq I \times \mathbb{R}$. Let $f: I \to \mathbb{R}$ be the function obtained by Lagrange polynomial interpolating the $k$ points above. Consider the sup-norm $$||f||_{(\infty, 2)} = \sup_{x \in I} |f(x)| + \sup_{x \in I} |f'(x)| + \sup_{x \in I} |f''(x)|$$
I think it would be extremely cumbersome to express $||f||_{(\infty, 2)}$ a function of the values $(x_1, y_1), \ldots, (x_k, y_k)$. In the alternative, I'm looking for relatively simple expressions for the tightest possible upper bound and lower bounds of $||f||_{(\infty, 2)}$ as functions of the values $(x_1, y_1), \ldots, (x_k, y_k)$.