On inequalities for norms of intermediate derivatives on a finite interval

Abstract

For functionsf which have an absolute continuous (n−1)th derivative on the interval [0, 1], it is proved that, in the case ofn>4, the inequality

$$\left\| {f^{(n - 2)} } \right\|_\infty \leqslant 4^{n - 2} (n - 1) ! \left\| f \right\|_\infty + \left\| {f^{(n)} } \right\|_\infty /2$$

holds with the exact constant 4ⁿ⁻²(n−1)!.