We derive global curvature estimates for closed, strictly star-shaped \((n-2)\) -convex hypersurfaces in warped product manifolds, which satisfy the prescribed \((n-2)\) -curvature equation with a general right-hand side. The proof can be readily adapted to establish curvature estimates for semi-convex solutions to the general k-curvature equation. Furthermore, it can also be used to prove the same estimates for k-convex solutions to the prescribed curvature measure type equations.
