In this paper, we study the locally constrained inverse curvature flow for hypersurfaces in the half-space with \(\theta \) -capillary boundary, which was recently introduced by Wang et al. (Math Ann 388:2121–2154, 2024). Assume that the initial hypersurface is strictly convex with the contact angle \(\theta \in (0,\pi /2].\) We prove that the solution of the flow remains to be strictly convex for \(t>0,\) exists for all positive time and converges smoothly to a spherical cap. As an application, we prove a complete family of Alexandrov–Fenchel inequalities for convex capillary hypersurfaces in the half-space with the contact angle \(\theta \in (0,\pi /2].\) Along the proof, we develop a new tensor maximum principle for parabolic equations on compact manifold with proper Neumann boundary condition.
