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 $\Omega \ne \mathbb{C}$, 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 $\Vert \omega {\Vert }_{\infty }={\sup }_{z\in \mathbb{D}}|\omega (z)|$. Let $\mathcal{C}(n)=\{(c_0,c_1,\dots ,c_n)\in {\mathbb{C}}^{n+1}:\text{there exists}\omega \in H_1^{\infty }(\mathbb{D})\text{satisfying}\omega (z)=c_0+c_{1}z+\cdots +c_n{z}^n+\cdots$ for $z\in \mathbb{D}\}$. For each fixed $z_0\in \mathbb{D}$, $j=-1,0,1,2,\dots$ and $c=(c_0,c_1,\dots ,c_n)\in \mathcal{C}(n)$, we use the Schur algorithm to determine the region of variability $V_{\Omega }^j(z_0,c)=\{{\int }_0^{z_0}{z}^j(g(z)-g(0))dz:g\in {\mathcal{F}}_{\Omega }\text{with}(P^{-1}\circ g)(z)=c_0+c_{1}z+\cdots +c_n{z}^n+\cdots \}$. We also show that for $z_0\in \mathbb{D}\backslash \{0\}$ 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 $\partial V_{\Omega }^j(z_0,c)$.

18 pages