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Volume 33, Issue 3
Some Sharpening and Generalizations of a Result of T. J. Rivlin

N. K. Govil & E. R. Nwaeze

Anal. Theory Appl., 33 (2017), pp. 219-228.

Published online: 2017-08

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  • Abstract

Let $p(z)=a_0+a_1z+a_2z^2+a_3z^3+\cdots+a_nz^n$ be a polynomial of degree $n$. Rivlin [12] proved that if $p(z)\neq 0$ in the unit disk, then for $0< r\leq 1,$ $${\max_{|z|=r}|p(z)|}\geq \Big(\dfrac{r+1}{2}\Big)^n{\max_{|z|=1}|p(z)|}.$$ In this paper, we prove a sharpening and generalization of this result and show by means of examples that for some polynomials our result can significantly improve the bound obtained by the Rivlin's Theorem.

  • AMS Subject Headings

15A18, 30C10, 30C15, 30A10

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{ATA-33-219, author = {}, title = {Some Sharpening and Generalizations of a Result of T. J. Rivlin}, journal = {Analysis in Theory and Applications}, year = {2017}, volume = {33}, number = {3}, pages = {219--228}, abstract = {

Let $p(z)=a_0+a_1z+a_2z^2+a_3z^3+\cdots+a_nz^n$ be a polynomial of degree $n$. Rivlin [12] proved that if $p(z)\neq 0$ in the unit disk, then for $0< r\leq 1,$ $${\max_{|z|=r}|p(z)|}\geq \Big(\dfrac{r+1}{2}\Big)^n{\max_{|z|=1}|p(z)|}.$$ In this paper, we prove a sharpening and generalization of this result and show by means of examples that for some polynomials our result can significantly improve the bound obtained by the Rivlin's Theorem.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2017.v33.n3.3}, url = {http://global-sci.org/intro/article_detail/ata/10513.html} }
TY - JOUR T1 - Some Sharpening and Generalizations of a Result of T. J. Rivlin JO - Analysis in Theory and Applications VL - 3 SP - 219 EP - 228 PY - 2017 DA - 2017/08 SN - 33 DO - http://doi.org/10.4208/ata.2017.v33.n3.3 UR - https://global-sci.org/intro/article_detail/ata/10513.html KW - Inequalities, polynomials, zeros. AB -

Let $p(z)=a_0+a_1z+a_2z^2+a_3z^3+\cdots+a_nz^n$ be a polynomial of degree $n$. Rivlin [12] proved that if $p(z)\neq 0$ in the unit disk, then for $0< r\leq 1,$ $${\max_{|z|=r}|p(z)|}\geq \Big(\dfrac{r+1}{2}\Big)^n{\max_{|z|=1}|p(z)|}.$$ In this paper, we prove a sharpening and generalization of this result and show by means of examples that for some polynomials our result can significantly improve the bound obtained by the Rivlin's Theorem.

N. K. Govil & E. R. Nwaeze. (1970). Some Sharpening and Generalizations of a Result of T. J. Rivlin. Analysis in Theory and Applications. 33 (3). 219-228. doi:10.4208/ata.2017.v33.n3.3
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