Variability regions for the \(n\)-th derivative of bounded analytic functions

Abstract

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 \(\gamma = (\gamma_0, \gamma_1 , \ldots , \gamma_{n-1}) \in {\mathbb D}^{n}\), let \({\mathcal H} (\gamma) = \{f \in {\mathcal H} : f (z_0) = \gamma_0,H^1f (z_0) = \gamma_1,\ldots ,H^{n-1}f (z_0) = \gamma_{n-1} \}\). 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 Dieudonn\'e'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.