On an operator preserving inequalities between polynomials

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 $$|B[P\circσ](z)-αB[P\circρ](z)+β\{(\frac{R+k}{k+r})^{n}-|α|\}B[P\circρ](z)| $$ on the maximum and minimum modulus of $|P(z)|$ on $|z|=k$ for arbitrary real or complex numbers $α,β\in\mathbb{C}$ with $|α|\leq 1,|β|\leq 1,R>r\geq k,$ $σ(z)=Rz,$ $ρ(z)=rz$ and establish certain sharp operator preserving inequalities between polynomials, from which a variety of interesting results follow as special cases.