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 M̄=I×λM. Based on the conjecture proposed by Ren-Wang in {Ren2}, which is valid for k≥ n-2, we derive curvature estimates for equation σₖ(κ)= ψ (V, ν (V)) through a straightforward proof. Furthermore, we also obtain an existence result for the star-shaped compact hypersurface Σ satisfying the above equation by the degree theory under some sufficient conditions.