  In this paper, we establish a Heintze-Karcher type inequality for
hypersurfaces with capillary boundary of contact angle $\theta\in
(0,\frac{\pi}{2})$ in a half space or a half ball, by using solution to a mixed
boundary value problem in Reilly type formula. Consequently, we give a new
proof of Alexandrov type theorem for embedded capillary constant mean curvature
hypersurfaces with contact angle $\theta\in (0,\frac{\pi}{2})$ in a half space
or a half ball.
