Article
Ufa Mathematical Journal
Volume 11, Number 1, pp. 114-120
On zeros of polynomial
Das S.
DOI:10.13108/2019-11-1-114
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For a given polynomial
\begin{equation*}
P\left( z\right) =z^{n}+a_{n-1}z^{n-1}+a_{n-2}z^{n-2}+\cdots +a_{1}z+a_{0}
\end{equation*}
with real or complex coefficients,
the Cauchy bound
\begin{equation*}
\left\vert z\right\vert <1+A,\qquad A=\underset{0\leqslant j\leqslant n-1}{%
\max }\left\vert a_{j}\right\vert
\end{equation*}
does not reflect the fact that for $A$ tending to zero, all the zeros of $P\left( z\right) $ approach the origin $z=0$. Moreover, Guggenheimer (1964)
generalized the Cauchy bound by using a lacunary type polynomial
\begin{equation*}
p\left( z\right) =z^{n}+a_{n-p}z^{n-p}+a_{n-p-1}z^{n-p-1}+\cdots
+a_{1}z+a_{0}, \qquad 0 < p < n\text{.}
\end{equation*}
In this paper we obtain new results related with above facts. Our first result is the best possible. For the case as $A$
tends to zero, it reflects the fact that all the zeros of P(z) approach the origin $z=0$; it also sharpens the result obtained by Guggenheimer.
The rest of the related results concern zero-free bounds giving
some important corollaries. In many cases the new bounds are much
better than other well-known bounds.