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

Abstract:

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 \[ F(z):=\int _{0}^{z}h(t)B(t)dt \] satisfies \[ |F’(z)|=\Phi (|F(z)|^2), \quad z\in \partial \mathbb {D}. \] 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.