Korean J. Math. Vol. 29 No. 4 (2021) pp.825-832
DOI: https://doi.org/10.11568/kjm.2021.29.4.825

# On Sendov's conjecture about critical points of a polynomial

## Abstract

The derivative of a polynomial $p(z)$ of degree $n$, with respect to point $\alpha$ is defined by $D_{\alpha}p(z)=np(z)+(\alpha-z)p'(z)$. Let $p(z)$ be a polynomial having all its zeros in the unit disk $|z| \leq 1$. The Sendov conjecture asserts that if all the zeros of a polynomial $p(z)$ lie in the closed unit disk, then there must be a zero of $p'(z)$ within unit distance of each zero. In this paper, we obtain certain results concerning the location of the zeros of $D_{\alpha}p(z)$ with respect to a specific zero of $p(z)$ and a stronger result than Sendov conjecture is obtained. Further, a result is obtained for zeros of higher derivatives of polynomials having multiple roots.

## References

 A. Aziz, On the location of critical points of Polynomials, J. Austral. Math. Soc. 36 (1984), 4–11. Google Scholar

 B. Bojanov, Q. I. Rahman and J Syznal, On a conjecture of Sendov about critical points of a polynomial Mathematische Zeitschrift, 190 (1985), 281–286. Google Scholar

 J. E. Brown and G. Xiang, Proof of Sendov conjecture for polynomials of degree at most eight, J. Math. Anal. Appl. 232 (1999), 272–292. Google Scholar

 A. W. Goodman, Q. I. Rahman and J. S. Ratti, On the zeros of polynomial and its derivative, Proc. Amer. Math. Soc. 21 (1969), 273–274. Google Scholar

 Q.I Rahman and G. Schmeisser, Analytic Theory of Polynomials, Oxford University Press, (2002). Google Scholar

 G. Schmeisser, Bemerkungen zu einer Vermutung von Ilief, Math. Z 111 (1969), 121–125. Google Scholar

 Tereence Tao, Sendov’s Conjecture for sufficiently high degree polynomials, arXiv:2012.04125v1. Google Scholar