@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 = {<p style="text-align: justify;">Let $p(z)=\sum^n_{v=0}a_vz^v$&nbsp;be a polynomial of degree $n$,<br/>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $M(p,R)=:\underset{|z|=R\geq 0}{\max}|p(z)|$ and $M(p,1)=:||p||$.<br/>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].</p>},
issn = {1573-8175},
doi = {https://doi.org/10.4208/ata.OA-2018-0001},
url = {http://global-sci.org/intro/article_detail/ata/17132.html}
}