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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.04030 |
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Table of Contents:
- 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.