Hiroshima Mathematical Journal

A quantitative result on Sendov's conjecture for a zero near the unit circle

Tomohiro Chijiwa

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Abstract

On Sendov's conjecture, V. Vâjâitu and A. Zaharescu (and M. J. Miller, independently) state the following in their paper: if one zero $a$ of a polynomial which has all the zeros in the closed unit disk is sufficiently close to the unit circle, then the distance from $a$ to the closest critical point is less than $1$. It is desirable to quantify this assertion. In the author's previous paper, we obtained an upper bound on the radius of the disk centered at the origin which contains all the critical points. In this paper, we improve it, and then, estimate the range of the zero $a$ satisfying the above. This result, moreover, implies that if a zero of a polynomial is close to the unit circle and all the critical points are far from the zero, then the polynomial must be close to $P(z)=z^n-c$ with $\abs{c}=1$.

Article information

Source
Hiroshima Math. J., Volume 41, Number 2 (2011), 235-273.

Dates
First available in Project Euclid: 24 August 2011

Permanent link to this document
https://projecteuclid.org/euclid.hmj/1314204564

Digital Object Identifier
doi:10.32917/hmj/1314204564

Mathematical Reviews number (MathSciNet)
MR2849157

Zentralblatt MATH identifier
1232.30008

Subjects
Primary: 12D10: Polynomials: location of zeros (algebraic theorems) {For the analytic theory, see 26C10, 30C15}
Secondary: 26C10, 30C15

Keywords
Sendov's conjecture polynomial zero critical point

Citation

Chijiwa, Tomohiro. A quantitative result on Sendov's conjecture for a zero near the unit circle. Hiroshima Math. J. 41 (2011), no. 2, 235--273. doi:10.32917/hmj/1314204564. https://projecteuclid.org/euclid.hmj/1314204564


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