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Ufa Mathematical Journal, 2019, Volume 11, Issue 1, Pages 114–120
DOI: https://doi.org/10.13108/2019-11-1-114
(Mi ufa465)
 

On zeros of polynomial

Subhasis Das

Department of Mathematics, Kurseong College, Dow Hill Road, 734203, Kurseong, India
References:
Abstract: 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.
Keywords: zeroes, region, Cauchy bound, Lacunary type polynomials.
Received: 30.08.2017
Bibliographic databases:
Document Type: Article
UDC: 512.622.2
MSC: 30C15, 30C10, 26C10
Language: English
Original paper language: English
Citation: Subhasis Das, “On zeros of polynomial”, Ufa Math. J., 11:1 (2019), 114–120
Citation in format AMSBIB
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\by Subhasis~Das
\paper On zeros of polynomial
\jour Ufa Math. J.
\yr 2019
\vol 11
\issue 1
\pages 114--120
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