We fix two non-zero natural integers $n$ and $m$ as well as two polynomials $Q \in \mathbb{C}[X]$ and $R \in \mathbb{C}[X]$ of degrees $n$ and $m$ respectively. We introduce the polynomial $P = QR$ and we denote by $\lambda$ its leading coefficient and $\alpha_1, \ldots, \alpha_{n+m}$ its roots counted with multiplicity. Show that there exist $u$ and $v$ in $\partial\mathbb{D}$ such that: $$\|Q\|_{\mathbb{D}} \|R\|_{\mathbb{D}} \leq |\lambda| \cdot \prod_{i=1}^{n+m} \max\left\{\left|u - \alpha_i\right|, \left|v - \alpha_i\right|\right\}.$$
We fix two non-zero natural integers $n$ and $m$ as well as two polynomials $Q \in \mathbb{C}[X]$ and $R \in \mathbb{C}[X]$ of degrees $n$ and $m$ respectively. We introduce the polynomial $P = QR$ and we denote by $\lambda$ its leading coefficient and $\alpha_1, \ldots, \alpha_{n+m}$ its roots counted with multiplicity.
Show that there exist $u$ and $v$ in $\partial\mathbb{D}$ such that:
$$\|Q\|_{\mathbb{D}} \|R\|_{\mathbb{D}} \leq |\lambda| \cdot \prod_{i=1}^{n+m} \max\left\{\left|u - \alpha_i\right|, \left|v - \alpha_i\right|\right\}.$$