Write $H_{f}$ for the Hessian of a real function $f:\mathbb{R}^n\mapsto \mathbb{R}$, and define the bordered Hessian as
H_{f} = \left(\begin{matrix}0 & \nabla f' \\ \nabla f & H \end{matrix}\right),\quad H := \left[ \frac{\partial f}{\partial x_ix_j}\right]_{i,j} \quad i,j=\overline{1,n}
and consider the composition $g = h \circ f$. What is the best way to show the relationship of $H_g$ and $H_f$?