Semnan UniversityInternational Journal of Nonlinear Analysis and Applications2008-682213120220301Inequalities for an operator on the space of polynomials431439551410.22075/ijnaa.2021.22378.2355ENNisar AhmadRatherDepartment of Mathematics,
University of KashmirAaqib IqbalDepartment of Mathematics,
University of KashmirIshfaq AhmadDarDepartment of Mathematics,
University of KashmirJournal Article20210113Let $\mathcal{P}_n$ 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 $N : \mathcal{P}_n\rightarrow \mathcal{P}_n$ <br />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.https://ijnaa.semnan.ac.ir/article_5514_783dd656b7da60f84dab87531da4e7eb.pdf