Generalized Ahlfors functions

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 ${\mathcal F}$.