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2011 A quantitative result on Sendov's conjecture for a zero near the unit circle
Tomohiro Chijiwa
Hiroshima Math. J. 41(2): 235-273 (2011). DOI: 10.32917/hmj/1314204564


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 $\abs{c}=1$.


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Tomohiro Chijiwa. "A quantitative result on Sendov's conjecture for a zero near the unit circle." Hiroshima Math. J. 41 (2) 235 - 273, 2011.


Published: 2011
First available in Project Euclid: 24 August 2011

zbMATH: 1232.30008
MathSciNet: MR2849157
Digital Object Identifier: 10.32917/hmj/1314204564

Primary: 12D10
Secondary: 26C10, 30C15

Keywords: critical point , polynomial , Sendov's conjecture , zero

Rights: Copyright © 2011 Hiroshima University, Mathematics Program


Vol.41 • No. 2 • 2011
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