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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A note on the location of critical points of polynomials


Authors: E. B. Saff and J. B. Twomey
Journal: Proc. Amer. Math. Soc. 27 (1971), 303-308
MSC: Primary 30.11
DOI: https://doi.org/10.1090/S0002-9939-1971-0271312-1
MathSciNet review: 0271312
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Abstract: Let $ \mathcal{P}(a,3)$ denote the set of cubic polynomials which have all of their zeros in $ \vert z\vert \leqq 1$ and at least one zero at $ z = a(\vert a\vert \leqq 1)$. In this paper we describe a minimal region $ \mathcal{D}(a,3)$ with the property that every polynomial in $ \mathcal{P}(a,3)$ has at least one critical point in $ \mathcal{D}(a,3)$. The location of the zeros of the logarithmic derivative of the function $ {(z - a)^m}{(z - {z_1})^{{m_1}}}{(z - {z_2})^{{m_2}}}$ is also discussed.


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DOI: https://doi.org/10.1090/S0002-9939-1971-0271312-1
Keywords: Critical points of polynomials, zeros of the derivative Ilieff conjecture, cubic polynomials, logarithmic derivatives of polynomials
Article copyright: © Copyright 1971 American Mathematical Society