Volume 36, Issue 2
On Sharpening of a Theorem of Ankeny and Rivlin

Anal. Theory Appl., 36 (2020), pp. 225-234.

Published online: 2020-06

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• 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].

15A18, 30C10, 30C15, 30A10

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@Article{ATA-36-225, author = {Dalal , Aseem and K. Govil , N.}, title = {On Sharpening of a Theorem of Ankeny and Rivlin}, journal = {Analysis in Theory and Applications}, year = {2020}, volume = {36}, number = {2}, pages = {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].

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-2018-0001}, url = {http://global-sci.org/intro/article_detail/ata/17132.html} }
TY - JOUR T1 - On Sharpening of a Theorem of Ankeny and Rivlin AU - Dalal , Aseem AU - K. Govil , N. JO - Analysis in Theory and Applications VL - 2 SP - 225 EP - 234 PY - 2020 DA - 2020/06 SN - 36 DO - http://doi.org/10.4208/ata.OA-2018-0001 UR - https://global-sci.org/intro/article_detail/ata/17132.html KW - Inequalities, polynomials, zeros. AB -

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].

Aseem Dalal & N. K. Govil. (2020). On Sharpening of a Theorem of Ankeny and Rivlin. Analysis in Theory and Applications. 36 (2). 225-234. doi:10.4208/ata.OA-2018-0001
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