Polynomial and rational inequalities on Jordan arcs and domains

TL;DR

This paper establishes a sharp Bernstein-type inequality for polynomials on Jordan arcs and explores domain mappings via rational functions, advancing understanding of polynomial inequalities on complex domains.

Contribution

It introduces a new asymptotically sharp Bernstein-type inequality for polynomials on Jordan arcs and presents a general domain mapping result using rational functions.

Findings

Proved a sharp Bernstein-type inequality for polynomials on Jordan arcs.

Developed a general mapping theorem for domains bounded by Jordan curves.

Connected polynomial inequalities with rational function mappings and classical estimates.

Abstract

In this paper we prove an asymptotically sharp Bernstein-type inequality for polynomials on analytic Jordan arcs. Also a general statement on mapping of a domain bounded by finitely many Jordan curves onto a complement to a system of the same number of arcs with rational function is presented here. This fact, as well as, Borwein-Erd\'elyi inequality for derivative of rational functions on the unit circle, Gonchar-Grigorjan estimate of the norm of holomorphic part of meromorphic functions and Totik's construction of fast decreasing polynomials play key roles in the proof of the main result.