Let $\mathcal{B}_{n}$ be the set of matrices $M$ in $\mathcal{M}_{n}(\mathbf{C})$ such that the sequence $\left(\left\|M^{k}\right\|_{\mathrm{op}}\right)_{k \in \mathbf{N}}$ is bounded. For $M \in \mathcal{B}_{n}$, we set $b(M) = \sup\left\{\left\|M^{k}\right\|_{\mathrm{op}}; k \in \mathbf{N}\right\}$. For $M \in \mathcal{B}_{n}$, we define the function $$\varphi_{M}: z \in \mathbf{C} \backslash \mathbb{D} \longmapsto (|z|-1)\left\|R_{z}(M)\right\|_{\mathrm{op}}.$$ Deduce from the previous question the inequality $$\forall M \in \mathcal{B}_{n}, \quad \forall z \in \mathbf{C} \backslash \mathbb{D}, \quad \varphi_{M}(z) \leq b(M)$$
Let $\mathcal{B}_{n}$ be the set of matrices $M$ in $\mathcal{M}_{n}(\mathbf{C})$ such that the sequence $\left(\left\|M^{k}\right\|_{\mathrm{op}}\right)_{k \in \mathbf{N}}$ is bounded. For $M \in \mathcal{B}_{n}$, we set $b(M) = \sup\left\{\left\|M^{k}\right\|_{\mathrm{op}}; k \in \mathbf{N}\right\}$. For $M \in \mathcal{B}_{n}$, we define the function
$$\varphi_{M}: z \in \mathbf{C} \backslash \mathbb{D} \longmapsto (|z|-1)\left\|R_{z}(M)\right\|_{\mathrm{op}}.$$
Deduce from the previous question the inequality
$$\forall M \in \mathcal{B}_{n}, \quad \forall z \in \mathbf{C} \backslash \mathbb{D}, \quad \varphi_{M}(z) \leq b(M)$$