Remarks on complete Weingarten hypersurfaces in the Euclidean space

In this paper, first we provide a classification result for a large class of complete rotational Weingarten hypersurfaces in the Euclidean space. Second, we prove sharp inequalities for the norm of the second fundamental form of a class of Weingarten hypersurfaces. We show that sharpness is attained by a cylinder of the Euclidean space.