Curvature estimates for minimal surfaces with total boundary curvature less than 4$\pi$

We establish a curvature estimate for classical minimal surfaces with total boundary curvature less than 4$\pi$. The main application is a bound on the genus of these surfaces depending solely on the geometry of the boundary curve. We also prove that the set of simple closed curves with total curvature less than $4\pi$ and which do not bound an orientable compact embedded minimal surface of genus greater than $g$, for any given $g$, is open in the $C^{2,\alpha}$ topology.