Variability regions for Schur class

Let \({\mathcal S}\) be the class of analytic functions \(f\) in the unit disk \({\mathbb D}\) with \(f({\mathbb D}) \subset \overline{\mathbb D}\). Fix pairwise distinct points \(z_1,\ldots,z_{n+1}\in \mathbb{D}\) and corresponding interpolation values \(w_1,\ldots,w_{n+1}\in \overline{\mathbb{D}}\). Suppose that \(f\in{\mathcal S}\) and \(f(z_j)=w_j\), \(j=1,\ldots,n+1\). Then for each fixed \(z \in {\mathbb D} \backslash \{z_1,\ldots,z_{n+1} \}\), we obtained a multi-point Schwarz-Pick Lemma, which determines the region of values of \(f(z)\). Using an improved Schur algorithm in terms of hyperbolic divided differences, we solve a Schur interpolation problem involving a fixed point together with the hyperbolic derivatives up to a certain order at the point, which leads to a new interpretation to a generalized Rogosinski's Lemma. For each fixed \(z_0 \in {\mathbb D}\), \(j=1,2, \ldots n\) and \(\gamma = (\gamma_0, \gamma_1 , \ldots , \gamma_n) \in {\mathbb D}^{n+1}\), denote by \(H^jf(z)\) the hyperbolic derivative of order \(j\) of \(f\) at the point \(z\in {\mathbb D}\), let \({\mathcal S} (\gamma) = \{f \in {\mathcal S} : f (z_0) = \gamma_0,H^1f (z_0) = \gamma_1,\ldots ,H^nf (z_0) = \gamma_n \}\). We determine the region of variability \(V(z, \gamma ) = \{ f(z) : f \in {\mathcal S} (\gamma) \}\) for \(z\in {\mathbb D} \backslash \{ z_0 \}\), which can be called "the generalized Rogosinski-Pick Lemma for higher-order hyperbolic derivatives".