Landau--Kolmogorov inequality revisited

Abstract

The Landau-Kolmogorov problem consists of finding the upper bound $M_k$ for the norm of intermediate derivative $|f^{(k)}|$, when the bounds $|f|\le M_0$ and $|f^{(n)}|\le M_n$, for the norms of the function and of its higher derivative, are given. Here, we consider the case of a finite interval, and when all the norms are the max-norms. Our interest to that particular case is motivated by the fact that there are good chances to add this case to a short list of Landau--Kolmogorov inequalities where a complete solution exists, i.e., a solution that covers all values of $n,k\in \mathbb{N}$ (and, for a finite interval, all values of $\sigma =M_n/M_0$). The main guideline here is Karlin's conjecture that says that, for all $n,k\in \mathbb{N}$ and all $\sigma >0$, the maximum of $|f^{(k)}|$ is attained by a certain Chebyshev or Zolotarev spline. So far, it has been proved only for small $n\ge 4$ with all $\sigma$, and for all $n$ with particular $\sigma ={\sigma }_n$. Here, we prove Karlin's conjecture in several further subcases: 1) all $n,k\in \mathbb{N}$ and all $0<\sigma \le {\sigma }_n$ 2) all $n\in \mathbb{N}$, all $\sigma >0$, with $k=1,2$ 3) all $\sigma >0$, with $n<10$ and $0<k<n$.