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.
