  The superradiant stability of a Kerr-Newman black hole and charged massive
scalar perturbation is investigated. We treat the black hole as a background
geometry and study the equation of motion of the scalar perturbation. From the
radial equation of motion, we derive the effective potential experienced by the
scalar perturbation. By a careful analysis of this effective potential, it is
found that when the inner and outer horizons of Kerr-Newman black hole satisfy
$\frac{r_-}{r_+}\leqslant\frac{1}{3}$ and the charge-to-mass ratios of scalar
perturbation and black hole satisfy $ \frac{q}{\mu }\frac{Q}{ M}>1 $, the
Kerr-Newman black hole and scalar perturbation system is superradiantly stable.

General Relativity and Quantum Cosmology