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

 

Boundary differential relations for holomorphic functions on the disc


Authors: Miran Černe and Matej Zajec
Journal: Proc. Amer. Math. Soc. 139 (2011), 473-484
MSC (2010): Primary 30E25, 35Q15
Published electronically: July 8, 2010
MathSciNet review: 2736330
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Abstract: The existence of solutions of boundary differential relations for holomorphic functions on the disc $ \Delta$ is considered. First we prove that for an arbitrary continuous positive function $ \Phi$ on the complex plane $ \mathbb{C}$ there exists a disc algebra function $ f\in A(\Delta)$ such that $ \vert f^{\prime}\vert=\Phi(f)$ on $ \partial\Delta$. Assuming some smoothness, the existence result is also proved for a quite general differential relation $ \rho(\xi,f^{\prime}(\xi))=\Phi(\xi, f(\xi))$, $ \xi\in\partial\Delta$, where $ \rho$ is a defining function for a family of Jordan curves in $ \mathbb{C}$ containing point 0 in its interior and $ \Phi$ is a bounded positive function on $ \partial\Delta\times\mathbb{C}$.


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Additional Information

Miran Černe
Affiliation: Department of Mathematics, University of Ljubljana, Jadranska 21, 1111 Ljubljana, Slovenia
Email: miran.cerne@fmf.uni-lj.si

Matej Zajec
Affiliation: Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1111 Ljubljana, Slovenia
Email: matej.zajec@imfm.uni-lj.si

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10469-0
PII: S 0002-9939(2010)10469-0
Keywords: Boundary value problem, Riemann-Hilbert problem
Received by editor(s): February 21, 2010
Received by editor(s) in revised form: March 1, 2010
Published electronically: July 8, 2010
Additional Notes: The first author was supported in part by grant Analiza in geometrija P1-0291 from the Ministry of Higher Education, Science and Technology of the Republic of Slovenia.
Communicated by: Franc Forstneric
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.