An operator product inequalities for polynomials

Abstract

Let $P(z)$ be a polynomial of degree $n\ge 1$. In this paper we define an operator $B$, as following, $$B[P(z)]:={\lambda }_{0}P(z)+{\lambda }_1(\frac{nz}{2})\frac{P^{\prime }(z)}{1!}+{\lambda }_2(\frac{nz}{2}{)}^2\frac{P^{\prime \prime }(z)}{2!},$$ where ${\lambda }_0,{\lambda }_1$ and ${\lambda }_2$ are such that all the zeros of $$u(z)={\lambda }_0+c(n,1){\lambda }_{1}z+c(n,2){\lambda }_2{z}^2$$ lie in half plane $$|z|\le |z-\frac{n}{2}|$$ and obtain a new generalization of some well-known results.

11 pages