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 $\varphi_{M}: z \in \mathbf{C} \backslash \mathbb{D} \longmapsto (|z|-1)\left\|R_{z}(M)\right\|_{\mathrm{op}}$. We fix $r \in ]1, +\infty[$ and $(X,Y) \in \Sigma_{n}^{2}$. For $\rho \in \mathbf{R}^{+*}$, we denote $\mathbb{D}_{\rho} = \{z \in \mathbf{C}; |z| \leq \rho\}$. By using the previous question, integration by parts and inequality (3) from question $17$, show that $$\forall k \in \mathbf{N}, \quad \left|X^{T}M^{k}Y\right| \leq \frac{r^{k+1}}{(k+1)(r-1)} n b^{\prime}(M).$$
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 $\varphi_{M}: z \in \mathbf{C} \backslash \mathbb{D} \longmapsto (|z|-1)\left\|R_{z}(M)\right\|_{\mathrm{op}}$.
We fix $r \in ]1, +\infty[$ and $(X,Y) \in \Sigma_{n}^{2}$. For $\rho \in \mathbf{R}^{+*}$, we denote $\mathbb{D}_{\rho} = \{z \in \mathbf{C}; |z| \leq \rho\}$.
By using the previous question, integration by parts and inequality (3) from question $17$, show that
$$\forall k \in \mathbf{N}, \quad \left|X^{T}M^{k}Y\right| \leq \frac{r^{k+1}}{(k+1)(r-1)} n b^{\prime}(M).$$