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\title{Abstract}
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 Let $\mathbb{D}=\{z\in\mathbb {C}:|z|\leq1\}$ be the closed unit disk in $\mathbb{C}$.\ \ Let $\mathbb{C}[z]$ denote the set of polynomials $P(z)=a_0z^n+a_1z^{n-1}+a_2z^{n-3}+a_{n-1}z+a_n,$  where $a_k\in\mathbb{C},$  $k\in\{0,1,2,\ldots,n\}$  \   and  \ $n\in\mathbb{N}^*.$\\
    We will study sufficient conditions regarding the roots of a polynomial $P\in\mathbb{C}[z]$  which imply the following  conjecture, attributed  to the bulgarian  mathematician  Blagovest Sendov.
    \begin{conjecture}
    If all the roots of a polynomial $P\in\mathbb{C}[z]$ lie in $\mathbb{D}$ and $z^*$ is an arbitrary root of the polynomial  $P$  then the disk $\{z\in{\mathbb{C}}: |z -z^*|\leq1\}$ contains at least one root of $P'$.
    \end{conjecture}
We will give sufficient conditions which imply the Sendov's
conjecture.
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