On an operator preserving inequalities between polynomials

Abstract

Let ${\mathcal{P}}_n$ 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 ${\mathcal{P}}_n$ 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 $|\alpha |\le 1,|\beta |\le 1,R>r\ge k,$ $\sigma (z)=Rz,$ $\rho (z)=rz$ and establish certain sharp operator preserving inequalities between polynomials, from which a variety of interesting results follow as special cases.

15 pages