On constrained Markov-Nikolskii type inequality for $k-$absolutely monotone polynomials

TL;DR

This paper investigates the bounds of derivatives of $k$-absolutely monotone polynomials, extending classical inequalities by determining exact orders for all $p,q$ norms and showing improvements when $q<p$.

Contribution

It provides the exact order of constrained Markov-Nikolskii inequalities for all $p,q$ values, including cases where $q<p$, improving previous bounds.

Findings

Exact order established for all $0<p,q extless=ty$.

Significant improvements found for the case $q<p$.

Extends classical inequalities to broader polynomial classes.

Abstract

We consider the classical problem of estimating norms of higher order derivatives of algebraic polynomial via the norms of polynomial itself. The corresponding extremal problem for general polynomials in uniform norm was solved by A. A. Markov. In $1926,$ Bernstein found the exact constant in the Markov inequality for monotone polynomials. T. Erdelyi showed that the order of the constants in constrained Markov-Nikolskii inequality for $k-$ absolutely monotone polynomials is the same as in the classical one in case $0<p\le q\le\infty.$ In this paper, we find the exact order for all values of $0<p,q\le\infty.$ It turned out that for the case $q<p$ constrained Markov-Nikolskii inequality can be significantly improved.