Given a compact Riemannian manifold M and an open interval I in ℝ, we consider a warped product manifold \(\bar{M}=I \times_{\phi} M\) . For a positive function f defined on \(\bar{M}\) , we obtain the existence of the (η, k)-convex hypersurface Σ which satisfies a Hessian quotient equation \({\sigma_{k}(\lambda(\eta))\over \sigma_{l}(\lambda(\eta))}=f(V, \nu(V))\) for 0 ⩽ l < k < n, and η = Hg − h, the first Newton transformation of the second fundamental form h. This generalizes the result of Chen et al. (2020) and gives a simpler proof of the curvature estimate. As a corollary, we can get an (η, k)-convex solution for \(f(V, \nu)= \langle V, \nu \rangle\vert V \vert^{-n-1}h({V\over \vert V \vert})\) in ℝn+1 for some prescribed function h, which can be viewed as the prescribed curvature measure type problem.
