A result related to the Sendov conjecture

The Sendov conjecture asserts that if $p(z)={\prod }_{j=1}^N(z-z_j)$ is a polynomial with zeros $\mathrm{\mid }{z}_j\mathrm{\mid }\le 1$, then each disk $\mathrm{\mid }z-z_j\mathrm{\mid }\le 1$ contains a zero of $p^{\mathrm{\prime }}$. Our purpose is the following: Given a zero $z_j$ of order $n\ge 2$, determine whether there exists $\zeta ≠z_j$ such that $p^{\mathrm{\prime }}(\zeta )=0$ and $\mathrm{\mid }{z}_j-\zeta \mathrm{\mid }\le 1$. In this paper we present some partial results on the problem.