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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Authors: Richard Fournier and Stephan Ruscheweyh
Journal: Proc. Amer. Math. Soc. 127 (1999), 3287-3294
MSC (1991): Primary 30C45; Secondary 30C55
Published electronically: May 11, 1999
MathSciNet review: 1618666
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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

\begin{displaymath}F(z):=\int _{0}^{z}h(t)B(t)dt \end{displaymath}

satisfies

\begin{displaymath}|F'(z)|=\Phi (|F(z)|^2), \quad z\in\partial {\mathbb D }. \end{displaymath}

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.


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Additional Information

Richard Fournier
Affiliation: Centre de Recherches de Mathématiques, Université de Montréal, Montréal, Canada H3C 3J7
Email: fournier@DMS.UMontreal.CA

Stephan Ruscheweyh
Affiliation: Mathematisches Institut, Universität Würzburg\ D-97074 Würzburg, Germany
Email: ruscheweyh@mathematik.uni-wuerzburg.de

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04960-6
PII: S 0002-9939(99)04960-6
Received by editor(s): February 2, 1998
Published electronically: May 11, 1999
Additional Notes: The first author acknowledges support of FCAR (Quebec).
Communicated by: Albert Baernstein II
Article copyright: © Copyright 1999 American Mathematical Society