Free boundary value problems for analytic functions in the closed unit disk

We shall prove (a slightly more general version of) the following theorem: let $\Phi$ be analytic in the closed unit disk $\overline{\mathbb{D}}$ with $\Phi:[0,1]\rightarrow (0,1]$, and let $B(z)$ be a finite Blaschke product. Then there exists a function $h$ satisfying: i) $h$ analytic in the closed unit disk $\overline{\mathbb{D}}$, ii) $h(0)>0$, iii) $h(z)\neq 0$ in $\overline{\mathbb{D}}$, such that

satisfies

This completes a recent result of Kühnau for $\Phi (x)=1+\alpha x$, $-1<\alpha<0$, where this boundary value problem has a geometrical interpretation, namely that $\beta (\alpha )F(r(\alpha )z)$ preserves hyperbolic arc length on $\partial\mathbb{D}$ for suitable $\beta (\alpha ),$
$r(\alpha )$. For these important choices of $\Phi$ we also prove that the corresponding functions $h$ are uniquely determined by $B$, and that $zh(z)$ is univalent in ${\mathbb D }$. Our work is related to Beurling's and Avhadiev's on conformal mappings solving free boundary value conditions in the unit disk.