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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Generalized Ahlfors functions


Authors: Miran Cerne and Manuel Flores
Journal: Trans. Amer. Math. Soc. 359 (2007), 671-686
MSC (2000): Primary 35Q15; Secondary 32E99, 30E25
Published electronically: July 20, 2006
MathSciNet review: 2255192
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Abstract: Let $ \Sigma$ be a bordered Riemann surface with genus $ g$ and $ m$ boundary components. Let $ \lbrace\gamma_{z}\rbrace_{z\in\partial\Sigma}$ be a smooth family of smooth Jordan curves in $ \mathbb{C}$ which all contain the point 0 in their interior. Let $ p\in\Sigma$ and let $ {\mathcal F}$ be the family of all bounded holomorphic functions $ f$ on $ \Sigma$ such that $ f(p)\ge 0$ and $ f(z)\in \widehat{\gamma_z}$ for almost every $ z\in\partial\Sigma$. Then there exists a smooth up to the boundary holomorphic function $ f_0\in {\mathcal F}$ with at most $ 2g+m-1$ zeros on $ \Sigma$ so that $ f_0(z)\in\gamma_z$ for every $ z\in\partial\Sigma$ and such that $ f_0(p)\ge f(p)$ for every $ f\in {\mathcal F}$. If, in addition, all the curves $ \lbrace\gamma_z\rbrace_{z\in\partial\Sigma}$ are strictly convex, then $ f_0$ is unique among all the functions from the family $ {\mathcal F}$.


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Additional Information

Miran Cerne
Affiliation: Department of Mathematics, University of Ljubljana, Jadranska 19, 1111 Ljubljana, Slovenia
Email: miran.cerne@fmf.uni-lj.si

Manuel Flores
Affiliation: Department of Mathematics, University of La Laguna, 38771 La Laguna, Tenerife, Spain
Email: mflores@ull.es

DOI: http://dx.doi.org/10.1090/S0002-9947-06-03906-7
PII: S 0002-9947(06)03906-7
Keywords: Bordered Riemann surface, Ahlfors function, Riemann-Hilbert problem
Received by editor(s): June 21, 2004
Received by editor(s) in revised form: November 22, 2004
Published electronically: July 20, 2006
Additional Notes: The first author was supported in part by a grant “Analiza in geometrija” P1-0291 from the Ministry of Education, Science and Sport of the Republic of Slovenia. Part of this work was done while the author was visiting the University of La Laguna, Tenerife, Spain. He wishes to thank the faculty of the Analysis Department for their hospitality and support.
The second author was supported in part by grants from FEDER y Ministerio de Ciencia y Tecnologia number BFM2001-3894 and Consejeria de Educacion Cultura y Deportes del Gobierno de Canarias, PI 2003/068
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.