On an operator preserving inequalities between polynomials

Let P(z) be a polynomial of degree at most n. We consider an operator N, which carries a polynomial P(z) into $$\begin{aligned} N[P](z):=\sum \limits _{j=0}^{m}\lambda _j\bigg (\frac{nz}{2}\bigg )^j\frac{P^{(j)}(z)}{j!}, \end{aligned}$$ where $$\lambda _0,\lambda _1,\ldots ,\lambda _m$$ are such that all the zeros of $$\begin{aligned} u(z)=\sum \limits _{j=0}^{m}\left( {\begin{array}{c}n\\ j\end{array}}\right) \lambda _jz^j \end{aligned}$$ lie in the half plane $$\begin{aligned} |z|\le \bigg |z-\frac{n}{2}\bigg |. \end{aligned}$$ In this paper, we estimate the minimum and maximum modulii of N[P(z)] on $$|z|=1$$ with restrictions on the zeros of P(z) and thereby obtain compact generalizations of some well known polynomial inequalities.