Variability regions for the $n$-th derivative of bounded analytic functions

Let $\mathcal{H}$ be the class of all analytic self-maps of the open unit disk $\mathbb{D}$. Denote by $H^{n}f(z)$ the $n$-th order hyperbolic derivative of $f\in \mathcal{H}$ at $z\in \mathbb{D}$. For $z_0\in \mathbb{D}$ and $\gamma =({\gamma }_0,{\gamma }_1,\dots ,{\gamma }_{n-1})\in {\mathbb{D}}^n$, let $\mathcal{H}(\gamma )=\{f\in \mathcal{H}:f(z_0)={\gamma }_0,H^{1}f(z_0)={\gamma }_1,\dots ,H^{n-1}f(z_0)={\gamma }_{n-1}\}$. In this paper, we determine the variability region $V(z_0,\gamma )=\{f^{(n)}(z_0):f\in \mathcal{H}(\gamma )\}$, which can be called ``the generalized Schwarz-Pick Lemma of $n$-th derivative". We then apply the generalized Schwarz-Pick Lemma to establish a $n$-th order Dieudonné’s Lemma, which provides an explicit description of the variability region $\{h^{(n)}(z_0):h\in \mathcal{H},h(0)=0,h(z_0)=w_0,h^{\mathrm{\prime }}(z_0)=w_1,\dots ,h^{(n-1)}(z_0)=w_{n-1}\}$ for given $z_0$, $w_0$, $w_1,\dots ,w_{n-1}$. Moreover, we determine the form of all extremal functions.