Global Curvature Estimates for $\sigma_k$ Curvature Equations with $k\geq n/2$
We establish a new concavity inequality for the elementary symmetric function \(\sigma_k\), which controls the spectral quadratic form arising from the second variation of \(\log\lambda_{\max}\). The proof is based on an easy--hard decomposition of the spectral variables. The easy region is treated using an optimal constrained concavity estimate for \(\sigma_k\), whereas the hard region is analyzed through G{\aa}rding-root coordinates and the concavity of the ordered partial sums of the inverse roots. As an application, for \(n/2\leq k<n\), we obtain global curvature estimates for closed star-shaped \(k\)-convex hypersurfaces satisfying \(\sigma_k(\kappa)=f(X,\nu)\) with a general positive right-hand side, together with the corresponding global-to-boundary estimates for Euclidean Hessian equations.