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Inequalities for an operator on the space of polynomials | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقاله 33، دوره 13، شماره 1، خرداد 2022، صفحه 431-439 اصل مقاله (319.24 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2021.22378.2355 | ||
| نویسندگان | ||
| Nisar Ahmad Rather1؛ Aaqib Iqbal1؛ Ishfaq Ahmad Dar* 2 | ||
| 1Department of Mathematics, University of Kashmir | ||
| 2Department of Mathematics, University of Kashmir | ||
| تاریخ دریافت: 24 دی 1399، تاریخ پذیرش: 23 فروردین 1400 | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| Polynomials؛ Operators؛ Inequalities in the complex domain | ||
| مراجع | ||
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[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. Milovanovic, D. S. Mitrinovic, Th. M. Rassias, Topics In Polynomials: Extremal Problems, Inequalities, Zeros, World Scientific Publications 1994. [9] G. Polya , G. Szego, 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. https://doi.org/10.1007/s11139-020-00261-2 [12] Q. I. Rahman , G. Schmeisser, Analytic theory of Polynomials, Clarendon Press Oxford 2002. | ||
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