Convexity of constant mean curvature graphs in $\mathbb{R}^{n+1}$ with planar boundary

We study the Dirichlet problem for a graph $\Sigma$ in $\mathbb{R}^{n+1}$ with normalized constant mean curvature $H>0$ and planar boundary $\Gamma=\partial \Omega$. Our main result is that the optimal solvability condition, namely that the normalized mean curvature $h$ of $\Gamma$ satisfies $h\geq H$, also suffices when $\Omega$ is strictly convex, to prove the strict convexity of $\Sigma$.