LAUSR

Let $$\Omega $$ Ω be a convex domain in the complex plane $${\mathbb C}$$ C with $$\Omega \not = {\mathbb C}$$ Ω ≠ C , and P be a conformal map of the unit disk $${\mathbb D}$$ D onto $$\Omega $$ Ω . Let $${\mathcal F}_\Omega $$ F Ω be the class of analytic functions g in $${\mathbb D}$$ D with $$g({\mathbb D}) \subset \Omega $$ g ( D ) ⊂ Ω . Also, let $$H_1^\infty ({\mathbb D})$$ H 1 ∞ ( D ) be the well known closed unit ball of the Banach space $$H^\infty ({\mathbb D})$$ H ∞ ( D ) of bounded analytic functions $$\omega $$ ω in $${\mathbb D}$$ D , with norm $$\Vert \omega \Vert _\infty = \sup _{z \in {\mathbb D}} |\omega (z)|$$ ‖ ω ‖ ∞ = sup z ∈ D | ω ( z ) | . Let $${\mathcal C}(n) = \{ (c_0,c_1 , \ldots , c_n ) \in {\mathbb C}^{n+1}: \text {there exists} \; \omega \in H_1^\infty ({\mathbb D}) \; \text {satisfying} \; \omega (z) = c_0+c_1z + \cdots + c_n z^n + \cdots ~\text {for} ~z\in \mathbb D\}$$ C ( n ) = { ( c 0 , c 1 , … , c n ) ∈ C n + 1 : there exists ω ∈ H 1 ∞ ( D ) satisfying ω ( z ) = c 0 + c 1 z + ⋯ + c n z n + ⋯ for z ∈ D } . For each fixed $$z_0 \in {\mathbb D}$$ z 0 ∈ D , $$j=-1,0,1,2, \ldots $$ j = - 1 , 0 , 1 , 2 , … and $$c = (c_0, c_1 , \ldots , c_n) \in {\mathcal C}(n)$$ c = ( c 0 , c 1 , … , c n ) ∈ C ( n ) , we use the Schur algorithm to determine the region of variability $$V_\Omega ^j (z_0, c ) = \{ \int _0^{z_0} z^{j}(g(z)-g(0))\, d z : g \in {\mathcal F}_\Omega \; \text {with} \; (P^{-1} \circ g) (z) = c_0 +c_1z + \cdots + c_n z^n + \cdots \}$$ V Ω j ( z 0 , c ) = { ∫ 0 z 0 z j ( g ( z ) - g ( 0 ) ) d z : g ∈ F Ω with ( P - 1 ∘ g ) ( z ) = c 0 + c 1 z + ⋯ + c n z n + ⋯ } . We also show that for $$z_0 \in {\mathbb D} \backslash \{ 0 \}$$ z 0 ∈ D \ { 0 } and $$c \in \text {Int} \, {\mathcal C}(n) $$ c ∈ Int C ( n ) , $$V_\Omega ^j (z_0, c )$$ V Ω j ( z 0 , c ) is a convex closed Jordan domain, which we determine by giving a parametric representation of the boundary curve $$\partial V_\Omega ^j (z_0, c )$$ ∂ V Ω j ( z 0 , c ) .