  The instability of rotating Kerr black holes due to massive scalar
perturbations is investigated. It is well known that a bosonic field impinging
on a Kerr black hole can be amplified as it scatters off the hole. This
superradiant scattering occurs for frequencies in the range $\omega<m\Omega$,
where $\Omega$ is the angular frequency of the black hole and $m$ is the
azimuthal harmonic index of the mode. If the incident field has a non-zero rest
mass, $\mu$, then the mass term effectively works as a mirror, reflecting the
scattered wave back towards the black hole. The wave may bounce back and forth
between the black hole and some turning point amplifying itself each time. This
may lead to a dynamical instability of the system, a phenomena known as a
"black-hole bomb". In this work we provide a bound on the instability regime of
rotating Kerr spacetimes. In particular, we show that Kerr black holes are
stable to massive perturbations in the regime $\mu\geq\sqrt{2}m\Omega$.
