Let $M \in \mathcal{B}_{n}$. We define
$$b^{\prime}(M) = \sup\left\{\varphi_{M}(z); z \in \mathbf{C} \backslash \mathbb{D}\right\}$$
where $\varphi_{M}: z \in \mathbf{C} \backslash \mathbb{D} \longmapsto (|z|-1)\left\|R_{z}(M)\right\|_{\mathrm{op}}$, and $b^{\prime}(M) \leq b(M)$.
Using the inequality from question 19, prove the inequality
$$(4) \quad b(M) \leq e n b^{\prime}(M).$$