Surjective isometries on a Banach space of analytic functions on the open unit disc

Let $H(\mathbb{D})$ be the linear space of all analytic functions on the open unit disc $\mathbb{D}$. We define $\mathcal{S}^\infty$ by the linear subspace of all $f \in H(\mathbb{D})$ with bounded derivative $f'$ on $\mathbb{D}$. We give the characterization of surjective, not necessarily linear, isometries on $\mathcal{S}^\infty$ with respect to the following two norms: $\| f \|_\infty + \| f' \|_\infty$ and $|f(a)| + \| f' \|_\infty$ for $a \in \mathbb{D}$, where $\| \cdot \|_\infty$ is the supremum norm on $\mathbb{D}$.