Convex hull theorem for multiply connected domains in the plane with an estimate of the quasiconformal constant

For any multiply connected domain $\Omega$ in $\mathbb{R}^2$, let $S$ be the boundary of the convex hull in $H^3$ of $\mathbb{R}^2 \backslash \Omega $ which faces $\Omega$. Suppose in addition that there exists a lower bound $l > 0$ of the hyperbolic lengths of closed geodesics in $\Omega$. Then there is always a $K$-quasiconformal mapping from $S$ to $\Omega$, which extends continuously to the identity on $\partial S = \partial \Omega$, where $K$ depends only on $l$. We also give a numerical estimate of $K$ by using the parameter $l$.