  In this article, we prove that if $(M,g)$ is a genus $G$ orientable surface
with a single boundary component $S^1$, and if $(D,g_0)$ is a disc such that
interior points are connected by unique geodesics and
  $$d_{(D,g_0)}(x,y) \geq d_{(M,g)}(x,y)$$ for all $x,y \in \partial M =
\partial D$, then $$(1 + \frac{2 G}{\pi}) \textrm{Area}(M,g) \geq
\textrm{Area}(D,g_0).$$
