Let $K$ be a closed, bounded and infinite subset of $\mathbb{C}$. We fix two non-zero natural integers $n$ and $m$, and we set: $$C_{n,m}^K = \sup\left\{\left.\frac{\|Q\|_K \|R\|_K}{\|QR\|_K}\right\rvert\, Q \in \mathbb{C}_n[X]\backslash\{0\}, R \in \mathbb{C}_m[X]\backslash\{0\}\right\} \in \mathbb{R} \cup \{+\infty\}$$ Deduce that there exist two monic polynomials $Q_1 \in \mathbb{C}_n[X]$ and $R_1 \in \mathbb{C}_m[X]$ such that: $$\frac{\left\|Q_1\right\|_K \left\|R_1\right\|_K}{\left\|Q_1 R_1\right\|_K} = C_{n,m}^K.$$
Let $K$ be a closed, bounded and infinite subset of $\mathbb{C}$. We fix two non-zero natural integers $n$ and $m$, and we set:
$$C_{n,m}^K = \sup\left\{\left.\frac{\|Q\|_K \|R\|_K}{\|QR\|_K}\right\rvert\, Q \in \mathbb{C}_n[X]\backslash\{0\}, R \in \mathbb{C}_m[X]\backslash\{0\}\right\} \in \mathbb{R} \cup \{+\infty\}$$
Deduce that there exist two monic polynomials $Q_1 \in \mathbb{C}_n[X]$ and $R_1 \in \mathbb{C}_m[X]$ such that:
$$\frac{\left\|Q_1\right\|_K \left\|R_1\right\|_K}{\left\|Q_1 R_1\right\|_K} = C_{n,m}^K.$$