Extremal problem of the norm of an intermediate derivative (Q810863)

The aim of the paper is to prove the following interesting result: Let \(M_ 0,M_ n>0\) and \(k\in \{1,2,...,n-1\}\) be given. Then the following problems are equivalent (i.e. \(\mu_ k=\lambda_ k):\) \[ \mu_ k=\sup \{\| f^{(k)}\|:\;f\in W^ n([0,1]),\quad \| f\| \leq M_ 0,\quad \| f^{(n)}\| \leq M_ n\},\quad and \] \[ \lambda_ k=\sup \{| f^{(k)}(0)|:\;f\in W^ n([0,1]),\quad \| f\| \leq M_ 0,\quad \| f^{(n)}\| \leq M_ n\}, \] where \(\| g\| = \sup_{t\in [0,1]}| g(t)|\). The proof of the theorem is very involved.