Generalized Ahlfors functions

Authors:
Miran Černe and Manuel Flores

Journal:
Trans. Amer. Math. Soc. **359** (2007), 671-686

MSC (2000):
Primary 35Q15; Secondary 32E99, 30E25

DOI:
https://doi.org/10.1090/S0002-9947-06-03906-7

Published electronically:
July 20, 2006

MathSciNet review:
2255192

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Abstract | References | Similar Articles | Additional Information

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 Černe**

Affiliation:
Department of Mathematics, University of Ljubljana, Jadranska 19, 1 111 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

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.