A special concavity property for positive Hessian quotient operators

We establish a special concavity property for positive Hessian quotient operators $\frac{{\sigma }_n(W)}{{\sigma }_{n-k}(W)},1\le k\le n-1$. As a consequence, we prove a Jacobi inequality for general symmetric tensor satisfying positive Hessian quotient equation on Riemannian manifolds.