A note on the location of critical points of polynomials

Saff, E. B.; Twomey, J. B.

doi:10.1090/S0002-9939-1971-0271312-1

A note on the location of critical points of polynomials
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by E. B. Saff and J. B. Twomey PDF

Proc. Amer. Math. Soc. 27 (1971), 303-308 Request permission

Abstract:

Let $\mathcal {P}(a,3)$ denote the set of cubic polynomials which have all of their zeros in $|z| \leqq 1$ and at least one zero at $z = a(|a| \leqq 1)$. 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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