  We verify that if $M$ is a compact minimal hypersurface in $\mathbb{S}^{n+1}$
whose squared length of the second fundamental form satisfying $0\leq
|A|^2-n\leq\frac{n}{22}$, then $|A|^2\equiv n$ and $M$ is a Clifford torus.
Moreover, we prove that if $M$ is a complete self-shrinker with polynomial
volume growth in $\mathbb{R}^{n+1}$ whose equation is given by (\ref{selfshr}),
and if the squared length of the second fundamental form of $M$ satisfies
$0\leq|A|^2-1\leq\frac{1}{21}$, then $|A|^2\equiv1$ and $M$ is a round sphere
or a cylinder. Our results improve the rigidity theorems due to Q. Ding and Y.
L. Xin \cite{DX1,DX2}.
