Starshaped compact hypersurfaces with prescribed Weingarten curvature in warped product manifolds

Given a compact Riemannian manifold M, we consider a warped product manifold \({\bar{M}} = I \times _h M\), where I is an open interval in \({\mathbb {R}}\). For a positive function \(\psi \) defined on \({\bar{M}}\), we generalize the arguments in Guan et al. (Commun. Pure Appl. Math. 68(8):1287–1325, 2015) and Ren and Wang (On the curvature estimates for Hessian equations, 2016. arXiv:1602.06535), to obtain the curvature estimates for Hessian equations \(\sigma _k(\kappa )=\psi (V,\nu (V))\). We also obtain some existence results for the starshaped compact hypersurface \(\Sigma \) satisfying the above equation with various assumptions.