Let
$M$ be an
$n$dimensional
closed hypersurface with constant mean curvature and constant scalar curvature in a unit sphere.
Denote by
$H$
and
$S$ the
mean curvature and the squared length of the second fundamental form. We prove that
if
$\alpha (n,H)\le S\le \alpha (n,H)+{C}_{n}{H}^{2}$, where
$n\ge 4$ and
$H\ne 0$, then
$S=\alpha (n,H)$ and
$M$ is a Clifford
torus
${\mathbb{\mathbb{S}}}^{n1}({r}_{1})\times {\mathbb{\mathbb{S}}}^{1}({r}_{2})$.
Here
$$\alpha (n,H)=n+\frac{{n}^{3}}{2(n1)}{H}^{2}\frac{n(n2)}{2(n1)}\sqrt{{n}^{2}{H}^{4}+4(n1){H}^{2}},$$ 
and
${C}_{n}$ is a positive constant
explicitly depending on
$n$.
The emphasis is that our gap theorem imposes no restriction on the range of mean
curvature. Moreover, we obtain gap theorems for complete hypersurfaces with
constant mean curvature and constant scalar curvature in space forms.
