## A quantitative result on Sendov’s conjecture for a zero near the unit circle.(English)Zbl 1232.30008

Let $$p$$ be a complex polynomial all of whose zeros lie in the closed unit disk. Concerning Sendov’s conjecture, it is known that if a zero $$a$$ of $$p$$ lies “sufficiently close” to the unit circle, then the distance from $$a$$ to the closest critical point is less than 1. In a previous paper, the author obtained an upper bound for the radius of the disk, centered at the origin, which contains all the critical points of $$p$$. In the paper under review, the author improves the aforementioned upper bound and then estimates the range of the zero $$a$$ such that the distance from $$a$$ to the closest critical point is less than 1. This result implies that if a zero of $$p$$ ($$\deg(p)=n$$), is close to the unit circle and all the critical points are far from the zero, then the polynomial $$p$$ must be close to $$z^n-c$$ with $$|c|=1$$.

### MSC:

 30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) 26C10 Real polynomials: location of zeros

### Keywords:

Sendov’s conjecture; critical point; zero of a polynomial
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### References:

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