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Abstract <p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(P(z)=\sum\limits_{j=0}^{n}a_{j}z^{j}\)</EquationSource> <!--ContMath2560044Ali-m1--> </InlineEquation> be a polynomial of degree <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(n\)</EquationSource> <!--ContMath2560044Ali-m2--> </InlineEquation> such that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(a_{n}\geq a_{n-1}\geq...\geq a_{1}\geq a_{0}\geq 0\)</EquationSource> <!--ContMath2560044Ali-m3--> </InlineEquation>. Then according to Eneström–Kakeya theorem all the zeros of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(P(z)\)</EquationSource> <!--ContMath2560044Ali-m4--> </InlineEquation> lie in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(|z|\leq 1\)</EquationSource> <!--ContMath2560044Ali-m5--> </InlineEquation>. This important result was recently extended to quaternionic setting by Carney, Gardner, Keaton, and Powers. In this paper, we shall generalize and refine the results of Carney, Gardner, Keaton, Powers, and Tripathi.</p>

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On the Zeros of Quaternionic Polynomials with Restricted Coefficient

  • 期刊论文

  • 发表日期：

    2026年4月18日

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  • M. Ali,
  • B. A. Zargar,
  • M. H. Gulzar

摘要

Abstract

Let \(P(z)=\sum\limits_{j=0}^{n}a_{j}z^{j}\) be a polynomial of degree \(n\) such that \(a_{n}\geq a_{n-1}\geq...\geq a_{1}\geq a_{0}\geq 0\) . Then according to Eneström–Kakeya theorem all the zeros of \(P(z)\) lie in \(|z|\leq 1\) . This important result was recently extended to quaternionic setting by Carney, Gardner, Keaton, and Powers. In this paper, we shall generalize and refine the results of Carney, Gardner, Keaton, Powers, and Tripathi.