A quantitative result on polynomials with zeros in the unit disk

Abstract

On Sendov's conjecture, M. J. Miller states the following in his paper [10,11]; if a zero $\beta$ of a polynomial which has all the zeros in the closed unit disk is sufficiently close to the unit circle, then the distance from $\beta$ to the closest critical point is less than or equal to 1. It is desirable to quantify this assertion. In this paper, we estimate the radius of the disk with center at 0 containing all the critical points and estimate the range of the zero $\beta$ satisfying the above for the first step. This result, moreover, implies that if Sendov’s conjecture is false, then the polynomial must be close to an extremal one.