On operators preserving inequalities between polynomials

TL;DR

This paper reviews how certain linear operators that preserve zeros also maintain Bernstein-type polynomial inequalities, highlighting their role in extremal polynomial properties.

Contribution

It demonstrates that zero-preserving linear operators inherently preserve Bernstein inequalities, connecting operator theory with polynomial extremal properties.

Findings

Zero-preserving operators maintain Bernstein inequalities

Operators mapping polynomials to polynomials preserve extremal properties

Theoretical link between zero-preservation and inequality preservation

Abstract

In this review paper, we explore operator aspects in extremal properties of Bernstein-type polynomial inequalities. We shall also see that a linear operator which send polynomials to polynomials and have zero-preserving property naturally preserve Bernstein's inequality.