Let $z \mapsto f(z) : = \sum_{n = 0}^N a_n z^n$ be a D4312: Complex polynomial function such that

(i)	\begin{equation} \forall \, z \in \mathbb{C} \left( f(z) = 0 \quad \implies \quad |z| \leq 1 \right) \end{equation}
(ii)	\begin{equation} \mathsf{Ker}(f) : = \{ z \in \mathbb{C} : f(z) = 0 \} \end{equation}

Then \begin{equation} \forall \, z \in \mathbb{C} \left( f'(z) = 0 \quad \implies \quad \exists \, z_0 \in \mathsf{Ker}(f) : |z - z_0| \leq 1 \right) \end{equation}

Let $z \mapsto f(z) : = \sum_{n = 0}^N a_n z^n$ be a D4312: Complex polynomial function such that

(i)	\begin{equation} \mathsf{Ker}(f) : = \{ z \in \mathbb{C} : f(z) = 0 \} \end{equation}
(ii)	\begin{equation} \mathsf{Ker}(f) \subseteq B[0, 1] \end{equation}

Then \begin{equation} \forall \, z \in \mathbb{C} \left( f'(z) = 0 \quad \implies \quad \exists \, z_0 \in \mathsf{Ker}(f) : z \in B[z_0, 1] \right) \end{equation}

Let $z \mapsto f(z) : = \sum_{n = 0}^N a_n z^n$ be a D4312: Complex polynomial function such that

(i)	\begin{equation} \mathsf{Ker}(f) : = \{ z \in \mathbb{C} : f(z) = 0 \} \end{equation}
(ii)	\begin{equation} \mathsf{Ker}(f') : = \{ z \in \mathbb{C} : f'(z) = 0 \} \end{equation}
(iii)	\begin{equation} \mathsf{Ker}(f) \subseteq B[0, 1] \end{equation}

Then \begin{equation} \forall \, z \in \mathbb{C} \left( z \in \mathsf{Ker}(f') \quad \implies \quad \exists \, z_0 \in \mathsf{Ker}(f) : z \in B[z_0, 1] \right) \end{equation}