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 $$|BP\circ\sigma-\alpha BP\circ\rho+\beta{(\frac{R+k}{k+r})^{n}-|\alpha|}BP\circ\rho| $$ 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.