  In this paper, we consider Weingarten curvature equations for $k$-convex
hypersurfaces with $n<2k$ in a warped product manifold
$\overline{M}=I\times_{\lambda}M$. Based on the conjecture proposed by Ren-Wang
in \cite{Ren2}, which is valid for $k\geq n-2$, 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 $\Sigma$ satisfying the above equation by the degree
theory under some sufficient conditions.
