A half-space type property in the Euclidean sphere

. We study the notion of strong r -stability for the context of closed hypersurfaces Σ n ( n ≥ 3) with constant ( r + 1)-th mean curvature H r +1 immersed into the Euclidean sphere S n +1 , where r ∈ { 1 ,...,n − 2 } . In this setting, under a suitable restriction on the r -th mean curvature H r , we establish that there are no r -strongly stable closed hypersurfaces immersed in a certain region of S n +1 , a region that is determined by a totally umbilical sphere of S n +1 . We also provide a rigidity result for such hypersurfaces.