Landau--Kolmogorov inequality revisited

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 $n,k\in\N$ (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 $n,k\in\N$ 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 $n,k\in\N$ and all $0 < \sigma \le \sigma_n$ 2) all $n \in \N$, all $\sigma > 0$, with $k=1,2$ 3) all $\sigma > 0$, with $n < 10$ and $0 < k < n$.