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: $$C_{n,m}^K = \sup\left\{\left.\frac{\|Q\|_K \|R\|_K}{\|QR\|_K}\right\rvert\, Q \in \mathbb{C}_n[X]\backslash\{0\}, R \in \mathbb{C}_m[X]\backslash\{0\}\right\} \in \mathbb{R} \cup \{+\infty\}$$ Show that $C_{n,m}^K > 1$. To do this, one may choose two distinct elements $a$ and $b$ in $K$ and verify that, for $\rho \in \mathbb{R}$ sufficiently large, we have $\left\|Q_\rho R_\rho\right\|_K < \left\|Q_\rho\right\|_K \left\|R_\rho\right\|_K$ with $Q_\rho(X) = X - (\rho(b-a)+a)$ and $R_\rho(X) = X - (\rho(a-b)+b)$.
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:
$$C_{n,m}^K = \sup\left\{\left.\frac{\|Q\|_K \|R\|_K}{\|QR\|_K}\right\rvert\, Q \in \mathbb{C}_n[X]\backslash\{0\}, R \in \mathbb{C}_m[X]\backslash\{0\}\right\} \in \mathbb{R} \cup \{+\infty\}$$
Show that $C_{n,m}^K > 1$.
To do this, one may choose two distinct elements $a$ and $b$ in $K$ and verify that, for $\rho \in \mathbb{R}$ sufficiently large, we have $\left\|Q_\rho R_\rho\right\|_K < \left\|Q_\rho\right\|_K \left\|R_\rho\right\|_K$ with $Q_\rho(X) = X - (\rho(b-a)+a)$ and $R_\rho(X) = X - (\rho(a-b)+b)$.