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Zbl 0586.30008
Zero-free regions for polynomials and some generalizations of Eneström- Kakeya theorem.
(English)
[J] Can. Math. Bull. 27, 265-272 (1984). ISSN 0008-4395; ISSN 1496-4287/e

Let $P(z)=a\sb nz\sp n+a\sb pz\sp p+...+a\sb 1z+a\sb 0\in \Subset [z]$ satisfy $a\sb pa\sb n\ne 0$ for some p with $0\le p<n$. By applying the Gershgorin theorem to a matrix which is similar to the companion matrix of P(z), the authors show that for each $r>0$ all zeros of P(z) lie in $\vert z\vert \le \max \{r,\sum\sp{p}\sb{j=0}\vert a\sb j/a\sb n\vert r\sp{j+1-n}\}$. From this the authors derive the following main results. Theorem 1. If for some $t>0$ the above P(z) satisfies $\vert a\sb n\vert \ge t\sp{n-j}\vert a\sb j\vert,0\le j\le p$, then all zeros of P(z) lie in $\vert z\vert \le K\sb 1/t$, where $K\sb 1(\ge 1)$ is the largest positive root of $K\sp{n+1}-K\sp n-K\sp p+1=0$. Theorem 4. If for some $t>0$ and some $k\in \{0,1,...,n\}$ P(z)$=\sum\sp{n}\sb{j=0}a\sb jz\sp j\in \Subset [z],n\in {\bbfN}$, satisfies $$t\sp n\vert a\sb n\vert \le t\sp{n-1}\vert a\sb{n-1}\vert \le...\le t\sp k\vert a\sb k\vert \ge t\sp{k-1}\vert a\sb{k-1}\vert \ge...\ge t\vert a\sb 1\vert \ge \vert a\sb 0\vert,$$ then all zeros of P(z) lie in $$\vert z\vert \le t\{(2t\sp k\vert a\sb k\vert /t\sp n\vert a\sb n\vert)-1\}+2\sum\sp{n}\sb{j=0}\vert a\sb j-\vert a\sb j\vert \vert /\vert a\sb n\vert t\sp{n-j-1}.$$ Next, by using Bernstein's inequality and a special case of Theorem 1, the authors derive Theorem 2. If P(z)$=\sum\sp{n}\sb{j=0}a\sb jz\sp j\in \Subset [z],a>0$ are given and if $\alpha\in {\bbfR}$ satisfies $\max\sb{\vert z\vert =a}\vert P(z)\vert =\vert P(ae\sp{i\alpha})\vert$, then P(z)$\ne 0$ holds for $\vert z-ae\sp{i\alpha}\vert <a/2n$. Finally, several special cases are discussed and generalizations of Theorem 4 are stated.
[H.-J.Runckel]
MSC 2000:
*30C15 Zeros of polynomials, etc. (one complex variable)
30C10 Polynomials (one complex variable)
30A10 Inequalities in the complex domain

Keywords: zero-free regions; Eneström-Kakeya theorem; Gershgorin theorem; Bernstein's inequality

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