Interior estimates for the Hessian quotient equations

In this paper, we establish interior $C^2$ estimates for admissible semiconvex solutions to the general Hessian quotient equation $\frac{{\sigma }_k}{{\sigma }_l}(D^{2}u)=f(x,u),$ for the cases $l=k-1$ and $l=k-2$, where $f$ is a positive $C^{1,1}$ function. Such estimates are known to fail in general for $k-l\ge 3$, even for convex solutions, as shown by counterexamples due to Lu. The main ingredient is a quantitative concavity inequality for the Hessian quotient operator under the semiconvex condition. Our result provides a unified argument to such general Hessian quotient for $2\le k\le n-1$ in arbitrary dimensions.