Extremal Problems of Bernstein-Type and an Operator Preserving Inequalities between Polynomials

Abstract Let $$\mathbb{P}_{n}$$ denote the space of all complex polynomials $$P(z)=\sum\limits_{j=0}^{n}a_{j}z^{j}$$ of degree at most $$n$$ . For any complex number $$\alpha,$$ $$D_{\alpha}P(z)=nP(z)+(\alpha-z)P^{\prime}(z)$$ , denote the polar derivative of the polynomial $$P(z)$$ with respect to $$\alpha$$ and $$N[P]$$ denote a family of operators that maps $$P\in\mathbb{P}_{n}$$ into itself. In this paper, we combine the operators $$N[P]$$ and $$D_{\alpha}$$ and establish certain operator preserving inequalities concerning polynomials, from which a variety of interesting results can be obtained as special cases.