Inequalities for an operator on the space of polynomials

Document Type : Research Paper


Department of Mathematics, University of Kashmir


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{}] 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.


[1] N.C. Ankeny , T.J. Rivlin, On a theorem of S. Bernstein, Pacific J. Math., 5 (1995), 849-852.
[2] Abdul Aziz, On the location of the zeros of certain composite polynomials, Pacific J. Math., 118(1985), no. 1, 17-26.
[3] A. Aziz , Q. M. Dawood, Inequalities for a polynomial and its derivative, J. Approx. Theory, 53 (1988), 155-162.
[4] A. Aziz , N. A. Rather, On an inequality of S. Bernstein and Gauss-Lucas Theorem, Analytic and Geometric Inequalities and Applications, Kluwer Academic Publishers, 1999, 29-35.
[5] S.Bernstein, Sur l’ordre de la meilleure approximation des fonctions continues par des polynˆomes de degr´e donn´e, Hayez, imprimeur des acad´emies royales, vol. 4, 1912.
[6] P. D. Lax, Proof of a conjecture of P. Erd¨os on the derivative of a polynomial, Bull. Amer. Math. Soc., 50(1994), no. 5, 509-513.
[7] M. Marden, Geometry of polynomials, Math Surveys, No. 3. Amer. Math. Soc. Providence 1949.
[8] G. V. Milovanovi´c, D. S. Mitrinovi´c , Th. M. Rassias, Topics In Polynomials: Extremal Problems, Inequalities, Zeros, World Scientific Publications 1994.
[9] G. P´olya , G. Szeg¨o, Aufgaben und lehrsatze aus der Analysis, Springer-Verlag,Berlin 1925.
[10] P. J. O’hara , R. S. Rodriguez, Some properties of self-inversive polynomials, Proc. Amer. Math. Soc., 44 (1974) 331-335.
[11] N.A. Rather, Ishfaq Dar , Suhail Gulzar, On the zeros of certain composite polynomials and an operator preserving inequalities, Ramanujan J., 54(2021) 605–612.
[12] Q. I. Rahman , G. Schmeisser, Analytic theory of Polynomials, Clarendon Press Oxford 2002.
Volume 13, Issue 1
March 2022
Pages 431-439
  • Receive Date: 13 January 2021
  • Accept Date: 12 April 2021