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

TL;DR

This paper demonstrates that sharp Kolmogorov-type inequalities for estimating derivatives' uniform norms can be derived from existing solutions to the Kolmogorov problem, extending the understanding of inequalities involving multiple derivatives.

Contribution

It shows that inequalities involving more than three norms can be obtained from known solutions to the Kolmogorov problem using standard techniques.

Findings

Derived sharp inequalities for multiple derivatives' norms.

Extended Kolmogorov inequalities to more than three norms.

Connected inequalities to existing Kolmogorov problem solutions.

Abstract

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