Superradiant stability of the Kerr black holes

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 $\mu \ge \sqrt{2}m{\mathrm{\Omega }}_H$, where $\mu$ is the proper mass of the scalar, $m$ is the azimuthal number of the scalar mode, and ${\mathrm{\Omega }}_H$ 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 $\mu <\sqrt{2}m{\mathrm{\Omega }}_H$, 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.