On estimates for the coefficients of a polynomial

Suhail Gulzar

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

Suhail Gulzar ¹

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

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

Résumé
Abstract

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.

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.

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 :

Suhail Gulzar ¹

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

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Suhail Gulzar. On estimates for the coefficients of a polynomial. Comptes Rendus. Mathématique, Volume 354 (2016) no. 4, pp. 357-363. doi : 10.1016/j.crma.2016.01.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.01.018/

Bibliographie
Cité par

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

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

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

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

M. Shafi; N. A. Rather; S. Gulzar A note on Visser’s inequality, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika (2024) no. 11, p. 51 | DOI:10.26907/0021-3446-2024-11-51-60
M. Shafi; N. A. Rather; S. Gulzar A Note on Visser’s Inequality, Russian Mathematics, Volume 68 (2024) no. 11, p. 44 | DOI:10.3103/s1066369x24700889
S. Gulzar; N. A. Rather; M. S. Wani On Visser’s Inequality Concerning an Estimation of Polynomial Coefficients, Russian Mathematics, Volume 66 (2022) no. 3, p. 9 | DOI:10.3103/s1066369x22030033
S. Gulzar; N. A. Rather On a Composition Preserving Inequalities between Polynomials, Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), Volume 53 (2018) no. 1, p. 21 | DOI:10.3103/s1068362318010041
S. Gulzar; N. A. Rather L p inequalities for the Schur–Szegő composition of polynomials, Acta Mathematica Hungarica, Volume 151 (2017) no. 1, p. 124 | DOI:10.1007/s10474-016-0668-0
S. Gulzar Some Zygmund type inequalities for the sth derivative of polynomials, Analysis Mathematica, Volume 42 (2016) no. 4, p. 339 | DOI:10.1007/s10476-016-0403-7

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