The case of equality in Landau's problem (Q2575980)

Summary: Kolmogorov (1949) determined the best possible constant \(K_{n, m}\) for the inequality \[ M_m (f) \leq K_{n, m}M^{(n-m)/n}_0 (f) M_n^{m/n}(f), \quad 0 < m < n, \] where \(f\) is any function with \(n\) bounded, piecewise continuous derivatives on \(\mathbb{R}\) and \(M_k(f) = \sup_{x \in \mathbb{R}}|f^{(k)}(x)|\). In this paper, we provide a relatively simple proof for the case of equality.