For a closed minimal immersed hypersurface M in \(\mathbb S^{n+1}\) with second fundamental form A, and each integer \(k\ge 2\) , define a constant \(\sigma _k=\dfrac{\int _M \left( \left|A\right|^2\right) ^k}{\left|M\right|}\) . We show that \(\sigma _k \ge 2^k\) provided \(n=2\) and M is not totally geodesic. When \(n=4\) and M has two distinct principal curvatures, we show \(\sigma _2 \ge 16\) . When \(n\ge 3\) and M has two distinct principal curvatures, for each integer \(k\ge 2\) , there exists a positive constant \(\delta _k(n)<n\) , if \(\left|A\right|^2\ge \delta _k(n)\) , we have \(\sigma _k\ge n^k\) . All the equality holds iff M is isometric to a Clifford hypersurface.
