A sharp integral inequality for compact Weingarten hypersurfaces under an Okumura type inequality

Our aim in this paper is to provide a sharp integral inequality involving the norm of the traceless second fundamental form of a wide class of compact (without boundary) linear Weingarten hypersurfaces (including those with two distinct principal curvatures) immersed into a Riemannian space form. In particular, we generalize the results of Alías Meléndez (2020) when the ambient space form is the unit Euclidean sphere and give a new estimate when the space form is either the Euclidean space or the hyperbolic space. The sharpness of our integral inequality is realized by the totally umbilical spheres and, when the ambient space is the unit Euclidean sphere, by Clifford tori.