Doubling Argument of the Hessian Estimate for the Hessian Quotient Equations

In this paper, we establish a doubling argument to obtain Hessian estimates for convex solutions to the Hessian quotient equation $\frac{σ_n}{σ_k}(D^2u) = f(x,u,Du)$ for $k=n-1$ and $k=n-2$ under the condition that $\log f$ is convex in the $Du$ variable. In particular, our approach is pointwise and