On estimates for the coefficients of a polynomial

Gulzar, Suhail

doi:10.1016/j.crma.2016.01.018

Complex analysis

On estimates for the coefficients of a polynomial
[Sur les bornes pour les coefficients d'un polynôme]

Présenté par : the Editorial Board

Gulzar, Suhail ¹

¹ Islamic University of Science and Technology, Kashmir, Awantipora 192122, India

Comptes Rendus. Mathématique, Tome 354 (2016) no. 4, pp. 357-363

Abstract (VO)
Résumé

If $P (z) = \sum_{j = 0}^{n} a_{j} z^{j}$ is a polynomial of degree n, then it was proved by Rahman and Schmeisser [4] that for every $p \in [0, + \infty]$ ,

| a_{n} | + | a_{0} | \leq 2 \frac{{‖ P ‖}_{p}}{{‖ 1 + z ‖}_{p}} .

In this paper, various estimates for the coefficients of a polynomial P are obtained which among other things include the above inequality as a special case.

Si $P (z) = \sum_{j = 0}^{n} a_{j} z^{j}$ est un polynôme de degré n, Rahman et Schmeisser [4] ont montré que, pour tout $p \in [0, + \infty]$ , on a

| a_{n} | + | a_{0} | \leq 2 \frac{{‖ P ‖}_{p}}{{‖ 1 + z ‖}_{p}} .

Nous obtenons ici diverses majorations pour les coefficients d'un polynôme qui, entre autres, incluent l'inégalité ci-dessus comme cas particulier.

Reçu le : 2015-10-19
Accepté le : 2016-01-26
Publié le : 2016-03-02

DOI : 10.1016/j.crma.2016.01.018

Affiliations des auteurs :

Gulzar, Suhail ¹

¹ Islamic University of Science and Technology, Kashmir, Awantipora 192122, India

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Gulzar, Suhail. On estimates for the coefficients of a polynomial. Comptes Rendus. Mathématique, Tome 354 (2016) no. 4, pp. 357-363. doi: 10.1016/j.crma.2016.01.018

Bibliographie
Cité par

[1] Arestov, V.V. On integral inequalities for trigonometric polynomials and their derivatives, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 45 (1981), pp. 3-22 (in Russian); English translation: Math. USSR, Izv., 18, 1982, pp. 1-17

[2] Gardner, R.B.; Govil, N.K. An $L_{p}$ inequality for a polynomial and its derivative, J. Math. Anal. Appl., Volume 193 (1995), pp. 490-496

[3] Rahman, Q.I.; Schmeisser, G. Analytic Theory of Polynomials, Oxford University Press, New York, 2002

[4] Rahman, Q.I.; Schmeisser, G. $L_{p}$ inequalities for polynomial, J. Approx. Theory, Volume 53 (1988), pp. 26-32

[5] Visser, C. A simple proof of certain inequalities concerning polynomials, Proc. K. Ned. Akad. Wet., Volume 47 (1945), pp. 276-281

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