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

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 $\abs{c}=1$.