Given $u \in C^2(\mathbb{D}, \mathbb{R})$ such that
$$u(x) \geq \frac{1}{1-||x||^2}$$
And let $\Omega$ be the set of points where the hessian matrix, $Hu \geq 0$. Prove or disprove that $Du: \Omega \to \mathbb{R}^2$ is surjective
It is pretty natural to see that $u|_{\Omega}$ is a convex function by the theorem:
Theorem: Given a convex region $D \in \mathbb{R}^n$ and function $f \in C^2(D, \mathbb{R})$, the epigraph is convex iff $H f \geq 0$ for $x \in D$.
How should I continue on this?