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

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

Published online: 2017-08

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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.

15A18, 30C10, 30C15, 30A10

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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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