Estimate for index of hypersurfaces in spheres with null higher order mean curvature

In a recent paper (Barros, Sousa in: Kodai Math. J. 2009) the authors proved that closed oriented non-totally geodesic minimal hypersurfaces of the Euclidean unit sphere have index of stability greater than or equal to n + 3 with equality occurring at only Clifford tori provided their second fundamental forms A satisfy the pinching: |A|² ≥ n. The natural generalization for this pinching is ?(r + 2)S _r+2 ≥ (n ? r)S _r  > 0. Under this condition we shall extend such result for closed oriented hypersurface Σ ⁿ of the Euclidean unit sphere ${\mathbb{S}^{n+1}}$ with null S _r+1 mean curvature by showing that the index of r-stability, ${Ind_{\Sigma^n}^{r}}$ , also satisfies ${Ind_{\Sigma^n}^{r}\ge n+3}$ . Instead of the previous hypothesis if we consider ${\frac{S_{r+2}}{{S_r}}}$ constant we have the same conclusion. Moreover, we shall prove that, up to Clifford tori, closed oriented hypersurfaces ${\Sigma^{n}\subset \mathbb{S}^{n+1}}$ with S _r+1 = 0 and S _r+2 < 0 have index of r-stability greater than or equal to 2n + 5.