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On Sharpening of a Theorem of Ankeny and Rivlin

On Sharpening of a Theorem of Ankeny and Rivlin
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Year:    2020

Author:    Aseem Dalal, N. K. Govil

Analysis in Theory and Applications, Vol. 36 (2020), Iss. 2 : pp. 225–234

Abstract

Let $p(z)=\sum^n_{v=0}a_vz^v$ be a polynomial of degree $n$,
                          $M(p,R)=:\underset{|z|=R\geq 0}{\max}|p(z)|$ and $M(p,1)=:||p||$.
Then according to a well-known result of Ankeny and Rivlin [1], we have for $R\geq 1$, $$M(p,R)\leq (\frac{R^n+1}{2})||p||.$$This inequality has been sharpened by Govil [4], who proved that for $R\geq 1$, $$M(p,R)\leq (\frac{R^n+1}{2})||p||-\frac{n}{2}(\frac{||p||^2-4|a_n|^2}{||p||})\left\{\frac{(R-1||p||)}{||p||+2|a_n|}-ln(1+\frac{(R-1)||p||}{||p||+2|a_n|})\right\}.$$In this paper, we sharpen the above inequality of Govil [4], which in turn sharpens the inequality of Ankeny and Rivlin [1].

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/ata.OA-2018-0001

Analysis in Theory and Applications, Vol. 36 (2020), Iss. 2 : pp. 225–234

Published online:    2020-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    10

Keywords:    Inequalities polynomials zeros.

Author Details

Aseem Dalal

N. K. Govil

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    Bibliography

    2022

    https://doi.org/10.1016/B978-0-12-811988-4.00014-1 [Citations: 0]