Short loops in surfaces with a circular boundary component

It is a classical theorem of Loewner that the systole of a Riemannian torus can be bounded in terms of its area. We answer a question of a similar flavor of Robert Young showing that if S is a Riemannian surface with connected boundary in Rn, such that the boundary curve is a standard unit circle, then the length of the shortest non-contractible loop in S is bounded in terms of the area of S.