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 $D_{\alpha }p(z)=np(z)+(\alpha -z)p^{\prime }(z)$. Let $p(z)$ be a polynomial having all its zeros in the unit disk $|z|\le 1$. 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 $p^{\prime }(z)$ 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.