On the maximum modulus of a polynomial and its polar derivative

Description

For a polynomial p ( z ) of degree n , having all zeros in | z | ≤ 1, Jain is shown that max | z | = 1 | D α t ⋯ D α 2 D α 1 p ( z ) | ≥ n ( n − 1 ) ⋯ ( n − t + 1 ) 2 t × [ { ( | α 1 | − 1 ) ⋯ ( | α t | − 1 ) } max | z | = 1 | p ( z ) | + { 2 t ( | α 1 | ⋯ | . For a polynomial p ( z ) of degree n , having all zeros in | z | ≤ 1, Jain is shown that max | z | = 1 | D α t ⋯ D α 2 D α 1 p ( z ) | ≥ n ( n − 1 ) ⋯ ( n − t + 1 ) 2 t × [ { ( | α 1 | − 1 ) ⋯ ( | α t | − 1 ) } max | z | = 1 | p ( z ) | + { 2 t ( | α 1 | ⋯ | .

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• Polynomial
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• Published by: biomed
• Published: 01 January 2011
• Reads: 33
• Language: English

Exrait

ZirehJournal of Inequalities and Applications2011,2011:111 http://www.journalofinequalitiesandapplications.com/content/2011/1/111

R E S E A R C H

On the maximum modulus its polar derivative

Ahmad Zireh

Correspondence: [e-mail] Department of Mathematics, Shahrood University of Technology, Shahrood, Iran

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Abstract For a polynomialp(z) of degreen, having all zeros in |z|≤1, Jain is shown that n(n−1)∙ ∙ ∙(n−t+ 1)   maxD α t ∙ ∙ ∙D α 2 D α 1 p(z)≥ × t 2 |z|=1  { ( | α 1 | − 1 ) ∙ ∙ ∙ ( | α t | − 1 ) } maxp(z) |z|=1   t  2 ( | α 1 | ∙ ∙ ∙ | α t | ) − { ( | α 1 | − 1 ) ∙ ∙ ∙ ( | α t | − 1 ) } min p(z) , |z|=1 | α 1 | ≥ 1, | α 2 | ≥ 1, ∙ ∙ ∙ | α t | ≥ 1,t<n.

In this paper, the above inequality is extended for the polynomials having all zeros in |z|≤k, wherek≤1. Our result generalizes certain wellknown polynomial inequalities. (2010) Mathematics Subject Classification. Primary 30A10; Secondary 30C10, 30D15. Keywords:Polar derivative, Polynomial, Inequality, Maximum modulus, Zeros

1. Introduction and statement of results Letp(z) be a polynomial of degreen, then according to the wellknown Bernstein’s inequality [1] on the derivative of a polynomial, we have   maxp(z)≤nmaxp(z) . (1:1) |z|=1|z|=1

This result is best possible and equality holding for a polynomial that has all zeros at the origin. If we restrict to the class of polynomials which have all zeros in |z|≤1, then it has been proved by Turan [2] that n    maxp(z)≥maxp(z) . (1:2) 2 |z|=1|z|=1

The inequality (1.2) is sharp and equality holds for a polynomial that has all zeros on |z| = 1. As an extension to (1.2), Malik [3] proved that ifp(z) has all zeros in |z|≤k, where k≤1, then

© 2011 Zireh; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution 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.