A note on the Kolmogorov-type inequalities for more than three norms

Abstract

In this note we show that sharp Kolmogorov-type inequalities that estimate the uniform norm $\Vert f^{(k)}\Vert$ of the $k$-th derivative of a function $f:\mathbb{R}\to \mathbb{R}$ by the values of the uniform norm of $f$ and uniform norms of several its higher derivatives ($\Vert f^{(r)}\Vert$ and $\Vert f^{(r-1)}\Vert$, or $\Vert f^{(r)}\Vert$ and $\Vert f^{(r-2)}\Vert$, or $\Vert f^{(r)}\Vert$, $\Vert f^{(r-1)}\Vert$ and $\Vert f^{(r-2)}\Vert$) using standard techniques can be obtained from the known solutions to the Kolmogorov problem about existence of a function with given norms of its derivatives.