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Generalized Ahlfors functions

Authors:	Miran Cerne Manuel Flores

Institution:	Department of Mathematics, University of Ljubljana, Jadranska 19, 1111 Ljubljana, Slovenia ; Department of Mathematics, University of La Laguna, 38771 La Laguna, Tenerife, Spain

Abstract:	Let $\Sigma$ be a bordered Riemann surface with genus and 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 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 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 is unique among all the functions from the family ${\mathcal F}$ .

Keywords:	Bordered Riemann surface Ahlfors function Riemann-Hilbert problem

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