  Let $M$ be an $n$-dimensional closed hypersurface with constant mean
curvature and constant scalar curvature in an unit sphere. Denote by $H$ and
$S$ the mean curvature and the squared length of the second fundamental form
respectively. We prove that if $S > \alpha (n, H)$, where $n\geq 4$ and $H\neq
0$, then $S > \alpha (n, H) + B_n\frac{n H^2}{n - 1}$. Here \[ \alpha (n, H) =
n + \frac{n^3}{2 (n - 1)} H^2 - \frac{n (n - 2)}{2 (n - 1)}\sqrt{n^2 H^4 + 4 (n
- 1) H^2}, \] $B_n=\frac{1}{5}$ for $4\leq n \leq 20$, and $B_n=\frac{49}{250}$
for $n>20$. Moreover, we obtain a gap theorem for complete hypersurfaces with
constant mean curvature and constant scalar curvature in space forms.
