We use the notation $R$ introduced in part I. Let $z \in \mathbb{C}$ such that $z^2 \neq 1$. We denote
$$r = \left|R\left(z^2 - 1\right)\right|, \quad s = \left|z + R\left(z^2 - 1\right)\right|, \quad t = \left|z - R\left(z^2 - 1\right)\right|, \quad h = \max(s,t)$$
We also denote $V_n(z) = U_{n+1}(z,-1)$ for all $z \in \mathbb{C}$ and $n \in \mathbb{N}$.\\
Prove that, for all $n \in \mathbb{N}$,
$$\left|V_n(z)\right| \leqslant \frac{h^{n+1}}{r}$$