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On the maximum modulus of a polynomial and its polar derivative

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9 pages
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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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: azireh@shahroodut.ac.ir Department of Mathematics, Shahrood University of Technology, Shahrood, Iran
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Open Access
polynomial
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Abstract For a polynomialp(z) of degreen, having all zeros in |z|1, Jain is shown that n(n1)∙ ∙ ∙(nt+ 1) maxDαt∙ ∙ ∙Dα2Dα1p(z)≥ × t 2 |z|=1 {(|α1| −1)∙ ∙ ∙(|αt| −1)}maxp(z) |z|=1 t 2(|α1| ∙ ∙ ∙ |αt|)− {(|α1| −1)∙ ∙ ∙(|αt| −1)}minp(z) , |z|=1 |α1| ≥1,|α2| ≥1,∙ ∙ ∙ |αt| ≥1,t<n.
In this paper, the above inequality is extended for the polynomials having all zeros in |z|k, wherek1. 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 Bernsteins 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 k1, 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.