Inequalities for a polynomial and its derivative
12 pages
English

Découvre YouScribe en t'inscrivant gratuitement

Je m'inscris

Inequalities for a polynomial and its derivative

-

Découvre YouScribe en t'inscrivant gratuitement

Je m'inscris
Obtenez un accès à la bibliothèque pour le consulter en ligne
En savoir plus
12 pages
English
Obtenez un accès à la bibliothèque pour le consulter en ligne
En savoir plus

Description

For a polynomial p ( z ) of degree n which has no zeros in | z | < 1 , Liman et al. (Appl. Math. Comput. 218:949-955, 2011) established for all α , β ∈ C with | α | ≤ 1 , | β | ≤ 1 , R > r ≥ 1 and | z | = 1 . In this paper, we extend the above inequality for the polynomials having no zeros in | z | < k , k ≤ 1 . Our result generalizes certain well-known polynomial inequalities. MSC: 30A10, 30C10, 30D15.

Sujets

Informations

Publié par
Publié le 01 janvier 2012
Nombre de lectures 12
Langue English

Extrait

Zireh Journal of Inequalities and Applications 2012, 2012 :210 http://www.journalonequalitiesandapplications.com/content/2012/1/210 R E S E A R C H Open Access Inequalities for a polynomial and its derivative Ahmad Zireh * * Correspondence: azireh@shahroodut.ac.ir ; Abstract azireh@gmail.com For a polynomial p ( z ) of degree n which has no zeros in | z | < 1, Liman et al. (Appl. Department of Mathematics, Shahrood University of Technology, Math. Comput. 218:949-955, 2011) established Shahrood, Iran p ( Rz ) – α p ( rz ) + β  rR ++11 n | α | p ( rz ) 12  R n α r n + β  rR ++11 n | α | r n + 1 – α + β  Rr ++11 n | α | | m z | = a 1 x p ( z ) R n α r n + β  rR ++11 n | α | r n 1 – α + β  Rr ++11 n | α | | m z | i = n 1 p ( z ) , for all α , β C with | α | ≤ 1, | β | ≤ 1, R > r 1 and | z | = 1. In this paper, we extend the above inequality for the polynomials having no zeros in | z | < k , k 1. Our result generalizes certain well-known polynomial inequalities. MSC: 30A10; 30C10; 30D15 Keywords: polynomial; inequality; maximum modulus; derivative; restricted zeros 1 Introduction and statement of results Let p ( z ) be a polynomial of degree n and p ( z ) be its derivative. Then it is well known that x | m z | a = p ( z ) n | m z | a = x p ( z ) , (.) and | z m | = a R > x p ( z ) R n | m z | a = x p ( z ) . (.) Inequality (.) is a famous result due to Bernstein [ ], whereas inequality (.) is a simple consequence of the maximum modulus principle (see [ ]). Both the above inequalities are sharp, and an equality in each holds for the polynomials having all their zeros at the origin. For the class of polynomials having no zeros in | z | < , inequalities (.) and (.) have respectively been replaced by max p ( z ) n max p ( z ) , (.) | z | = | z | = © 2012 Zireh; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu-tion License (http://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
  • Univers Univers
  • Ebooks Ebooks
  • Livres audio Livres audio
  • Presse Presse
  • Podcasts Podcasts
  • BD BD
  • Documents Documents