M.Ahmadi Baseri, M. Bidkham and Madjid Eshaghi: A generating operator of inequalities for polynomials, p.151-162

Let

be a polynomial of degree $n\geq 1$ . In this paper we consider an operator

, which carries a polynomial

into

$\begin{displaymath}B[P(z)]:=\lambda_0 P(z)+\lambda_1 (\frac{nz}{2}) \frac{P'(z)}{1!}+\lambda_2 (\frac{nz}{2})^2 \frac{P''(z)}{2!},\end{displaymath}$

where $\lambda_0,\lambda_1$ and $\lambda_2$ are such that all the zeros of

$\begin{displaymath}u(z)=\lambda_0 +c(n,1)\lambda_1 z+c(n,2) \lambda_2 z^2\end{displaymath}$

lie in half plane

$\begin{displaymath}\vert z\vert\leq \vert z-\frac{n}{2}\vert,\end{displaymath}$

and obtain new generalizations of some well-known results.