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

In this note we show that sharp Kolmogorov-type inequalities that estimate the uniform norm $\mathrm{\parallel }{f}^{(k)}\mathrm{\parallel }$ of the $k$-th derivative of a function $f\text{\hspace{0.17em}⁣}:\mathbb{R}\to \mathbb{R}$ by the values of the uniform norm of $f$ and uniform norms of several its higher derivatives ($\mathrm{\parallel }{f}^{(r)}\mathrm{\parallel }$ and $\mathrm{\parallel }{f}^{(r-1)}\mathrm{\parallel }$, or $\mathrm{\parallel }{f}^{(r)}\mathrm{\parallel }$ and $\mathrm{\parallel }{f}^{(r-2)}\mathrm{\parallel }$, or $\mathrm{\parallel }{f}^{(r)}\mathrm{\parallel }$, $\mathrm{\parallel }{f}^{(r-1)}\mathrm{\parallel }$ and $\mathrm{\parallel }{f}^{(r-2)}\mathrm{\parallel }$) 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.