  We study the superradiant stability of the system of a Kerr black hole and a
massive scalar perturbation. It was proved previously that this system is
superradiantly stable when $\mu\geq \sqrt{2}m\Omega_H$, where $\mu$ is the
proper mass of the scalar, $m$ is the azimuthal number of the scalar mode, and
$\Omega_H$ is the angular velocity of the Kerr black hole horizon. Our study is
a complementary work of this result. We analytically prove that in the
complementary parameter region $\mu<\sqrt{2}m\Omega_H$, when the parameters of
scalar perturbation and Kerr black hole satisfy two simple inequalities,
$\omega<\frac{\mu}{\sqrt{2}}$,~ $\frac{r_-}{r_+}<0.802$, the system is also
superradiantly stable.
