Is there any inequality that relates the norm of the determinant of the Hessian Matrix and the norm of the Laplacian in $L^2(\Omega)$?

Question

Let $\psi$ $\in H_0^2(\Omega)$ with $\Omega\subset \mathbb{R}^n$ bounded, simply connected and smooth. Is there any inequality that relates the norm of the determinant of the Hessian Matrix and the norm of the Laplacian in $L^2(\Omega)$? Any help would be deeply appreciated.