Hessian Estimates for Convex Solutions of Hessian Quotient Equations via Compactness

We give a compactness proof of interior Hessian estimates for convex solutions of ${\sigma }_n(D^{2}u)\mathrm{/}{\sigma }_{n-2}(D^{2}u)=1$ and ${\sigma }_n(D^{2}u)\mathrm{/}{\sigma }_{n-1}(D^{2}u)=f(x,u,Du)>0$, with $1\mathrm{/}f$ concave in the gradient variable in the latter case, revisiting results of Lu and Fung.