V. Markov's problem for monotone polynomials

TL;DR

This paper extends Markov's classical problem to monotone polynomials, providing exact constants in Bernstein inequalities and advancing the understanding of polynomial derivative bounds in this specific class.

Contribution

It solves the extremal problem for monotone polynomials in the uniform norm, deriving exact constants for Bernstein inequalities.

Findings

Exact constant in Bernstein inequality for monotone polynomials

Solution to Markov's extremal problem for monotone polynomials

Enhanced bounds for derivatives of monotone polynomials

Abstract

We consider the classical problem of estimating norm of the derivative of algebraic polynomial via the norm of polynomial itself. The corresponding extremal problem for general polynomials in uniform norm was solved by V. Markov. In this note we solve analogous problem for monotone polynomials. As a consequence, we find exact constant in Bernstein inequality for monotone polynomials.