$k$-convex hypersurfaces with prescribed Weingarten curvature in warped product manifolds

In this paper, we consider Weingarten curvature equations for $k$-convex hypersurfaces with $n<2k$ in a warped product manifold $\bar{M}=I{\times }_{\lambda }M$. Based on the conjecture proposed by Ren-Wang in \cite{Ren2}, which is valid for $k\ge n-2$, we derive curvature estimates for equation ${\sigma }_k(\kappa )=\psi (V,\nu (V))$ through a straightforward proof. Furthermore, we also obtain an existence result for the star-shaped compact hypersurface $\mathrm{\Sigma }$ satisfying the above equation by the degree theory under some sufficient conditions.