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 $γ= (γ_0, γ_1 , \ldots , γ_{n-1}) \in {\mathbb D}^{n}$, let ${\mathcal H} (γ) = \{f \in {\mathcal H} : f (z_0) = γ_0,H^1f (z_0) = γ_1,\ldots ,H^{n-1}f (z_0) = γ_{n-1} \}$. In this paper, we determine the variability region $V(z_0, γ) = \{ f^{(n)}(z_0) : f \in {\mathcal H} (γ) \}$, 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 $\{h^{(n)}(z_0): h\in \mathcal{H}, h(0)=0,h(z_0) =w_0, h'(z_0)=w_1,\ldots, h^{(n-1)}(z_0)=w_{n-1}\}$ for given $z_0$, $w_0$, $w_1,\dots,w_{n-1}$. Moreover, we determine the form of all extremal functions.
Let $z_0$ and $w_0$ be given points in the open unit disk $\mathbb{D}$ with $|w_0| < |z_0|$, and $\mathcal{H}_0$ be the class of all analytic self-maps $f$ of $\mathbb{D}$ normalized by $f(0)=0$. In this paper, we establish the fourth-order Dieudonné's Lemma and apply it to determine the variability region $\{f^{(4)}(z_0): f\in \mathcal{H}_0,f(z_0) =w_0, f'(z_0)=w_1, f''(z_0)=w_2\}$ for given $z_0,w_0,w_1,w_2$ and give the form of all the extremal functions.
We continue our study on variability regions in \cite{Ali-Vasudevarao-Yanagihara-2018}, where the authors determined the region of variability $V_Ω^j (z_0, c ) = \{ \int_0^{z_0} z^{j}(g(z)-g(0))\, d z : g({\mathbb D}) \subset Ω, \; (P^{-1} \circ g) (z) = c_0 +c_1z + \cdots + c_n z^n + \cdots \}$ for each fixed $z_0 \in {\mathbb D}$, $j=-1,0,1,2, \ldots$ and $c = (c_0, c_1 , \ldots , c_n) \in \mathbb{C}^{n+1}$, when $Ω\subsetneq\mathbb{C}$ is a convex domain, and $P$ is a conformal map of the unit disk ${\mathbb D}$ onto $Ω$. In the present article, we first show that in the case $n=0$, $j=-1$ …
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