%0 Journal Article
%T Inequalities for an operator on the space of polynomials
%J International Journal of Nonlinear Analysis and Applications
%I Semnan University
%Z 2008-6822
%A Rather, Nisar Ahmad
%A Iqbal, Aaqib
%A Dar, Ishfaq Ahmad
%D 2022
%\ 03/01/2022
%V 13
%N 1
%P 431-439
%! Inequalities for an operator on the space of polynomials
%K polynomials
%K Operators
%K Inequalities in the complex domain
%R 10.22075/ijnaa.2021.22378.2355
%X 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 $N : \mathcal{P}_n\rightarrow \mathcal{P}_n$ defined by $N[P](z):=\sum_{j=0}^{k}\lambda_j\left(\frac{nz}{2}\right)^j\frac{P^{(j)}(z)}{j!}, ~ k \leq n$ where $\lambda_j\in\mathbb{C}$, $j=0,1,2,\ldots,k$ are such that all the zeros of $\phi(z) = \sum_{j=0}^{k} \binom{n}{j}\lambda_j z^j$ lie in the half plane $|z| \leq \left| z - \frac{n}{2}\right|$ and established certain sharp Bernstein-type polynomial inequalities. In this paper, we prove some more general results concerning the operator $N : \mathcal{P}_n \rightarrow \mathcal{P}_n$ 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.
%U https://ijnaa.semnan.ac.ir/article_5514_783dd656b7da60f84dab87531da4e7eb.pdf