  We prove the validity of an inequality involving a mean of the area and the
length of the boundary of immersed disks whose boundaries are homotopically
non-trivial curves in an oriented compact manifold which possesses convex mean
curvature boundary, positive escalar curvature and admits a map to
$\mathbb{D}^2\times T^{n}$ with nonzero degree, where $\mathbb{D}^2$ is a disk
and $T^n$ is an $n$-dimensional torus. We also prove a rigidity result for the
equality case when the boundary is totally geodesic. This can be viewed as a
partial generalization of a result due to Lucas Ambr\'ozio in \cite{AMB} to
higher dimensions.
