Document Type : Research Paper

Authors

• Nisar Ahmad Rather
• Aaqib Iqbal
• Ishfaq Ahmad Dar

Department of Mathematics, University of Kashmir

Abstract

Let ${\mathcal{P}}_n$ be the class of all complex polynomials of degree at most $n.$ Recently Rather et. al.[ \On the zeros of certain composite polynomials and an operator preserving inequalities, Ramanujan J., 54(2021)  605–612. \url{https://doi.org/10.1007/s11139-020-00261-2}] introduced an operator $N:{\mathcal{P}}_n\to {\mathcal{P}}_n$
defined by $N[P](z):=\sum _{j=0}^k{\lambda }_j{\left(\frac{nz}{2}\right)}^j\frac{P^{(j)}(z)}{j!},k\le n$ where ${\lambda }_j\in \mathbb{C}$, $j=0,1,2,\dots ,k$ are such that all the zeros of $\varphi (z)=\sum _{j=0}^k(\genfrac{}{}{0ex}{}{n}{j}){\lambda }_j{z}^j$ lie in the half plane $|z|\le |z-\frac{n}{2}|$ and established certain sharp Bernstein-type polynomial inequalities. In this paper, we prove some more general results concerning the operator $N:{\mathcal{P}}_n\to {\mathcal{P}}_n$ preserving inequalities between polynomials. Our results not only contain several well known results as special cases but also yield certain new interesting results as special cases.

Keywords

• Polynomials
• Operators
• Inequalities in the complex domain