Let $K$ be a closed, bounded and infinite subset of $\mathbb{C}$. Show that, if the inequality $\|Q\|_K \|R\|_K \geq \|QR\|_K$ is an equality, then there exists $z_0 \in K$ such that:
$$\left|Q\left(z_0\right)\right| = \|Q\|_K, \quad \left|R\left(z_0\right)\right| = \|R\|_K \quad \text{and} \quad \left|Q\left(z_0\right)R\left(z_0\right)\right| = \|QR\|_K.$$