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 $|z_j| \leq 1$, then each disk $|z-z_j| \leq 1$ contains a zero of $p’$. Our purpose is the following: Given a zero $z_j$ of order $n \geq 2$, determine whether there exists $\zeta \ne z_j$ such that $p’(\zeta ) = 0$ and $|z_j - \zeta | \leq 1$. In this paper we present some partial results on the problem.