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

Let $\mathrm{\Omega }$ be a convex domain in the complex plane $\mathbb{C}$ with $\mathrm{\Omega }≠\mathbb{C}$, and $P$ be a conformal map of the unit disk $\mathbb{D}$ onto $\mathrm{\Omega }$. Let ${\mathcal{F}}_{\mathrm{\Omega }}$ be the class of analytic functions $g$ in $\mathbb{D}$ with $g(\mathbb{D})\subset \mathrm{\Omega }$, and $H_1^{\mathrm{\infty }}(\mathbb{D})$ be the closed unit ball of the Banach space $H^{\mathrm{\infty }}(\mathbb{D})$ of bounded analytic functions $\omega$ in ${\mathbb D}$,withnorm$| \omega |\infty = \sup{z \in {\mathbb D}} |\omega (z)|$. 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 $z_0\in \mathbb{D}$, $j=-1,0,1,2,\dots$ 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 }$\mathrm{.}Wealsoshowthatfor$z_0 \in {\mathbb D} \backslash { 0 }$and$c \in $\text{Int}$ , {\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 $\partial V_\Omega^j (z_0, c )$.