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 \(g({\mathbb D}) \subset \Omega \). Also, let \(H_1^\infty ({\mathbb D})\) be the well known 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 \({\mathcal C}(n) = \{ (c_0,c_1 , \ldots , c_n ) \in {\mathbb C}^{n+1}: \text {there exists} \; \omega \in H_1^\infty ({\mathbb D}) \; \text {satisfying} \; \omega (z) = c_0+c_1z + \cdots + c_n z^n + \cdots ~\text {for} ~z\in \mathbb D\}\). 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 \(c \in \text {Int} \, {\mathcal C}(n) \), \(V_\Omega ^j (z_0, c )\) is a convex closed Jordan domain, which we determine by giving a parametric representation of the boundary curve \(\partial V_\Omega ^j (z_0, c )\).

令\（\ Omega \）是复平面\（{\ mathbb C} \）中具有\（\ Omega \ not = {\ mathbb C} \）的凸域，而P是单位圆盘的共形映射\（{\ mathbb D} \）到\（\ Omega \）上。让\（{\ mathcal F} _ \欧米茄\）是类的解析函数克在\（{\ mathbb d} \）与\（克（{\ mathbb d}）\子集\欧米茄\） 。另外，令\（H_1 ^ \ infty（{\ mathbb D}）\）是有界解析函数\ {的Banach空间\（H ^ \ infty（{\ mathbb D}）\）的众所周知的闭合单位球\欧米加\）在\（{\ mathbb d} \），使用范数\（\ Vert \ omega \ Vert _ \ infty = \ sup _ {z \ in {\ mathbb D}} | \ omega（z）| \）。令\（{\ mathcal C}（n）= \ {（c_0，c_1，\ ldots，c_n）\ in {\ mathbb C} ^ {n + 1}：\ text {存在} \; \ omega \ in H_1 ^ \ infty（{\ mathbb D}）\; \ text {satisfying} \; \ omega（z）= c_0 + c_1z + \ cdots + c_n z ^ n + \ cdots〜\ text {for}〜z \ in \ mathbb D \} \）。对于每个固定的\（z_0 \ in {\ mathbb D} \），\（j = -1,0,1,2，\ ldots \）和\（c =（c_0，c_1，\ ldots，c_n）\ in {\ mathcal C}（n）\），我们使用Schur算法确定变异性区域\（V_ \ Omega ^ j（z_0，c）= \ {\ int _0 ^ {z_0} z ^ {j}（ g（z）-g（0））\，dz：g \ in {\数学F} _ \ Omega \; \ text {with} \;（P ^ {-1} \ circ g）（z）= c_0 + c_1z + \ cdots + c_n z ^ n + \ cdots \} \）。我们还显示了对于\（z_0 \ in {\ mathbb D} \反斜杠\ {0 \} \）和\（c \ in \ text {Int} \，{\ mathcal C}（n）\），\（ V_ \ Omega ^ j（z_0，c）\）是凸封闭的Jordan域，我们通过给出边界曲线\（\ partial V_ \ Omega ^ j（z_0，c）\）的参数表示来确定。