An application of the Schur algorithm to variability regions of certain analytic functions

Abstract

Let $\Omega$ be a convex domain in the complex plane ${\mathbb C}$ with $\Omega \not= {\mathbb C}$, and $P$ be a conformal map of the unit disk ${\mathbb D}$ onto $\Omega$. Let ${\mathcal F}_\Omega$ be the class of analytic functions $g$ in ${\mathbb D}$ with $g({\mathbb D}) \subset \Omega$, and $H_1^\infty ({\mathbb D})$ be the closed unit ball of the Banach space $H^\infty ({\mathbb D})$ of bounded analytic functions $\omega$ in ${\mathbb D}$, with norm $\| \omega \|_\infty = \sup_{z \in {\mathbb D}} |\omega (z)|$. Let ${\mathcal C}(n) = \{ (c_0,c_1 , \ldots , c_n ) \in {\mathbb C}^{n+1}: \text{there exists} \; \omega \in H_1^\infty ({\mathbb D}) \; \text{satisfying} \; \omega (z) = c_0+c_1z + \cdots + c_n z^n + \cdots$ for ${z\in \mathbb D}\}$. For each fixed $z_0 \in {\mathbb D}$, $j=-1,0,1,2, \ldots$ and $c = (c_0, c_1 , \ldots , c_n) \in {\mathcal C}(n)$, we use the Schur algorithm to determine the region of variability $V_\Omega^j (z_0, c ) = \{ \int_0^{z_0} z^{j}(g(z)-g(0))\, d z : g \in {\mathcal F}_\Omega \; \text{with} \; (P^{-1} \circ g) (z) = c_0 +c_1z + \cdots + c_n z^n + \cdots \}$. We also show that for $z_0 \in {\mathbb D} \backslash \{ 0 \}$ and $c \in \textrm{Int} \, {\mathcal C}(n) $, $V_\Omega^j (z_0, c )$ is a convex closed Jordan domain, which we determine by giving a parametric representation of the boundary curve $\partial V_\Omega^j (z_0, c )$.