On an operator preserving inequalities between polynomials

TL;DR

This paper investigates inequalities involving a family of operators acting on polynomials, establishing sharp bounds that depend on the polynomial's maximum and minimum modulus on a circle, generalizing previous results.

Contribution

It introduces new sharp inequalities for polynomial operators that depend on maximum and minimum modulus, extending classical polynomial inequality results.

Findings

Established sharp operator inequalities for polynomials

Derived bounds depending on maximum and minimum modulus on a circle

Unified various classical inequalities as special cases

Abstract

Let $\mathscr{P}_n $ denote the space of all complex polynomials $P(z)=\sum_{j=0}^{n}a_{j}{z}^{j}$ of degree $n$ and $\mathcal{B}_n$ a family of operators that maps $\mathscr{P}_n$ into itself. In this paper, we consider a problem of investigating the dependence of $$|B[P\circ\sigma](z)-\alpha B[P\circ\rho](z)+\beta\{(\frac{R+k}{k+r})^{n}-|\alpha|\}B[P\circ\rho](z)| $$ on the maximum and minimum modulus of $|P(z)|$ on $|z|=k$ for arbitrary real or complex numbers $\alpha,\beta\in\mathbb{C}$ with $|\alpha|\leq 1,|\beta|\leq 1,R>r\geq k,$ $\sigma(z)=Rz,$ $\rho(z)=rz$ and establish certain sharp operator preserving inequalities between polynomials, from which a variety of interesting results follow as special cases.