Variability regions for Schur class

Abstract

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".