Annular bounds for the zeros of a polynomial from companion matrix
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866916354147221504 |
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| author | Bhunia, Pintu Paul, Kallol |
| author_facet | Bhunia, Pintu Paul, Kallol |
| contents | Let $p(z)=z^n+a_{n-1}z^{n-1}+a_{n-2}z^{n-2}+\ldots+a_1z+a_0$ be a complex polynomial with $a_0\neq 0$ and $n\geq 3$. Several new upper bounds for the moduli of the zeros of $p$ are developed. In particular, if $α=\sqrt{\sum_{j=0}^{n-1}|a_j|^2}$ and $z$ is any zero of $p$, then we show that \begin{eqnarray*}
|z|^2 &\leq & \cos^2 \fracπ{n+1}+|a_{n-2}|+ \frac{1}{4} \left ( |a_{n-1}|+ { α} \right)^2 + \frac{1}{2}\sqrt{α^2-|a_{n-1}|^2} + \frac{1}{2}α, \end{eqnarray*} which is sharper than the Abu-Omar and Kittaneh's bound \begin{eqnarray*}
|z|^2 &\leq & \cos^2 \fracπ{n+1}+ \frac{1}{4} \left ( |a_{n-1}|+ { α}\right)^2 + α \end{eqnarray*} if and only if $2|a_{n-2}|< \sqrt{\sum_{j=0}^{n-1}|a_j|^2}-\sqrt{\sum_{j=0}^{n-2}|a_j|^2}. $ The upper bounds obtained here enable us to describe smaller annuli in the complex plane containing all the zeros of $p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_01334 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Annular bounds for the zeros of a polynomial from companion matrix Bhunia, Pintu Paul, Kallol Complex Variables Functional Analysis 26C10, 15A60 Let $p(z)=z^n+a_{n-1}z^{n-1}+a_{n-2}z^{n-2}+\ldots+a_1z+a_0$ be a complex polynomial with $a_0\neq 0$ and $n\geq 3$. Several new upper bounds for the moduli of the zeros of $p$ are developed. In particular, if $α=\sqrt{\sum_{j=0}^{n-1}|a_j|^2}$ and $z$ is any zero of $p$, then we show that \begin{eqnarray*} |z|^2 &\leq & \cos^2 \fracπ{n+1}+|a_{n-2}|+ \frac{1}{4} \left ( |a_{n-1}|+ { α} \right)^2 + \frac{1}{2}\sqrt{α^2-|a_{n-1}|^2} + \frac{1}{2}α, \end{eqnarray*} which is sharper than the Abu-Omar and Kittaneh's bound \begin{eqnarray*} |z|^2 &\leq & \cos^2 \fracπ{n+1}+ \frac{1}{4} \left ( |a_{n-1}|+ { α}\right)^2 + α \end{eqnarray*} if and only if $2|a_{n-2}|< \sqrt{\sum_{j=0}^{n-1}|a_j|^2}-\sqrt{\sum_{j=0}^{n-2}|a_j|^2}. $ The upper bounds obtained here enable us to describe smaller annuli in the complex plane containing all the zeros of $p$. |
| title | Annular bounds for the zeros of a polynomial from companion matrix |
| topic | Complex Variables Functional Analysis 26C10, 15A60 |
| url | https://arxiv.org/abs/2107.01334 |