  Let $\Sigma$ be a smooth closed hypersurface with non-negative Ricci
curvature, isometrically immersed in a space form. It has been proved in
\cite{P}, \cite{CZ}, and \cite{C2} that there are some $L^2$ inequalities on
$\Sigma$ which measure the stability of closed umbilical hypersurfaces or more
generally, closed hypersurfaces with traceless Newton transformation of the
second fundamental form. In this paper, we prove that the constants in these
inequalities are optimal.
