# Higher Markov and Bernstein inequalities and fast decreasing polynomials with prescribed zeros

## TL;DR

This paper develops advanced Bernstein- and Markov-type inequalities for specific classes of polynomials on certain compact sets, extending classical bounds and providing new tools for approximation theory.

## Contribution

It introduces higher order inequalities for trigonometric and algebraic polynomials with prescribed zeros, under specific geometric conditions.

## Key findings

- Established higher Markov and Bernstein inequalities for polynomials
- Extended inequalities to polynomials with prescribed zeros
- Applied results to compact subsets satisfying interval conditions

## Abstract

Higher order Bernstein- and Markov-type inequalities are established for trigonometric polynomials on compact subsets of the real line and algebraic polynomials on compact subsets of the unit circle. In the case of Markov-type inequalities we assume that the compact set satisfies an interval condition.