## 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
Full Text:

### References:

 [1] I. Borcea, The Sendov conjecture for polynomials with at most seven distinct zeros, Analysis 16 (1996), no. 2, 137-159. · Zbl 0859.30005 [2] D. A. Brannan, On a conjecture of Ilieff, Proc. Camb. Phil. Soc. 64 (1968), 83-85. · Zbl 0163.30901 [3] J. E. Brown, On the Sendov conjecture for sixth degree polynomials, Proc. Amer. Math. Soc. 113 (1991), no. 4, 939-946. · Zbl 0738.30007 [4] J. E. Brown and G. Xiang, Proof of the Sendov conjecture for polynomials of degree at most eight, J. Math. Anal. Appl. 232 (1999), no. 2, 272-292. · Zbl 0927.30006 [5] T. Chijiwa, A quantitative result on polynomials with zeros in the unit disk, Proc. Japan Acad. Ser. A Math. Sci. 86 (2010), no. 10, 165-168. · Zbl 1220.30006 [6] J. Dieudonné, Sur quelques applications de la théorie des fonctions bornées aux polynômes dont toutes les racines sont dans un domain circulaire donné, Actualités Scient. et Industri. 114 (1934), 5-24. · JFM 60.0291.02 [7] W. K. Hayman, Research Problems in Function Theory, Athlone Press University of London, London, 1967. · Zbl 0158.06301 [8] S. Kumar and B. G. Shenoy, On the Ilyeff-Sendov conjecture for polynomials with at most five zeros, J. Math. Anal. Appl. 171 (1992), 595-600. · Zbl 0773.30004 [9] M. Marden, Geometry of Polynomials, Second edition. Mathematical Surveys, no. 3, Amer. Math. Soc., Providence, RI, 1966. · Zbl 0162.37101 [10] A. Meir and A. Sharma, On Ilieff’s conjecture, Pacific J. Math. 31 (1969), no. 2, 459-467. · Zbl 0194.10202 [11] M. J. Miller, On Sendov’s conjecture for roots near the unit circle, J. Math. Anal. Appl. 175 (1993), no. 2, 632-639. · Zbl 0782.30007 [12] M. J. Miller, A quadratic approximation to the Sendov radius near the unit circle, Trans. Amer. Math. Soc. 357 (2004), no. 3, 851-873. · Zbl 1066.30007 [13] D. Phelps and R. S. Rodriguez, Some properties of extremal polynomials for the Ilieff Conjecture, Kōdai Math. Sem. Report 24 (1972), 172-175. · Zbl 0244.30001 [14] Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, London Mathematical Society Monographs. New Series, 26, Oxford Univ. Press, Oxford, 2002. · Zbl 1072.30006 [15] Z. Rubinstein, On a problem of Ilyeff, Pacific J. Math. 26 (1968), 159-161. · Zbl 0172.35902 [16] V. Vâjâitu and A. Zaharescu, Ilyeff’s conjecture on a corona, Bull. Lond. Math. Soc. 25 (1993), no. 3, 49-54. · Zbl 0796.30004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.