A concavity inequality and interior $C^2$ estimate for Hessian quotient equations

We establish a concavity inequality for the Hessian quotient operators $\frac{{\sigma }_k}{{\sigma }_l}$ in the cases $k-l\in \{1,2\}$, and then derive the corresponding Jacobi inequality. Combining this with the framework developed by Lu and Tsai, we obtain an interior Hessian estimate for convex solutions of $\frac{{\sigma }_k(D^{2}u)}{{\sigma }_l(D^{2}u)}=f$. As an application, we prove that any entire convex solution in ${\mathbb{R}}^n$ with quadratic growth must be a quadratic polynomial.