Inequalities for Lorentz polynomials

TL;DR

This paper establishes new inequalities for Lorentz polynomials, including a sharp Markov-type inequality, expanding the theoretical understanding of polynomial bounds with specific coefficient and zero constraints.

Contribution

It introduces a novel sharp Markov-type inequality for Lorentz polynomials with real coefficients and non-vanishing derivatives inside the unit disk.

Findings

Proves Nikolskii-type inequalities for Lorentz polynomials.

Derives a sharp Markov-type inequality for polynomials with specific zero conditions.

Compares new inequalities with classical results like Erdos's inequality.

Abstract

We prove a few interesting inequalities for Lorentz polynomials including Nikolskii-type inequalities. A highlight of the paper is a sharp Markov-type inequality for polynomials of degree at most n with real coefficients and with derivative not vanishing in the open unit disk. The result may be compared with Erdos's classical Markov-type inequality (1940) for polynomials of degree at most n having only real zeros outside the interval (-1,1).