Starshaped compact hypersurfaces with prescribed Weingarten curvature in warped product manifolds

Given a compact Riemannian manifold M, we consider awarped productmanifold (M) over bar = Ix(h)M, where I is an open interval in R. For a positive function psi defined on (M) over bar, wegeneralize the arguments in Guan et al. (Commun. Pure Appl. Math. 68(8):1287-1325, 2015) and Ren andWang (On the curvature estimates for Hessian equations, 2016. arXiv: 1602.06535), to obtain the curvature estimates for Hessian equations sigma(k)(kappa) =psi(V, v(V)). We also obtain some existence results for the starshaped compact hypersurface Sigma satisfying the above equation with various assumptions.