The following conjecture is well-known: Chern's conjecture. For \(n\)-dimensional closed minimal hypersurfaces in the unit sphere \(S^{n+1}(1)\) with constant scalar curvature, the values \(S\) of the squared norm of the second fundamental forms should be discrete. Relating to this conjecture, we prove the following main theorem: Theorem 1. Let \(M^n\) \((n>3)\) be a closed minimal hypersurface of the unit sphere \(S^{n+ 1}(1)\) with constant scalar curvature and \(S\) the squared norm of the second fundamental form. If \(S>n\), then \[ S- n\geq \textstyle{{1\over 3}} n. \] In the case that \(\sum_{i,j,k} h_{ij}h_{jk} h_{ki}= \sum_i\lambda^3_i=\text{const.}\), we also prove the following sharper estimate: Theorem 2. Let \(M^n\) \((n>3)\) be a closed minimal hypersurface of the unit sphere \(S^{n+1}(1)\) with constant scalar curvature and \(S\) the squared norm of the second fundamental form. Assume that \(\sum_{i,j,k} h_{ij} h_{jk} h_{ki}\equiv \sum_i\lambda^3_i= \text{const.}\), where \(II= \sum_{i,j} h_{ij}\omega_i \omega_j\) is the second fundamental form on \(M^n\). If \(S>n\), then \[ S- n\geq\textstyle{{2\over 3}} n. \]