  Spinning Kerr black holes are known to be superradiantly unstable to massive
scalar perturbations. We here prove that the instability regime of the composed
Kerr-black-hole-massive-scalar-field system is bounded from above by the
dimensionless inequality $M\mu < m \cdot \sqrt{{{2(1+\gamma)
(1-\sqrt{1-\gamma^2}) - \gamma^2} \over {4\gamma^2}}}$, where $\{\mu,m\}$ are
respectively the proper mass and azimuthal harmonic index of the scalar field
and $\gamma\equiv r_-/r_+$ is the dimensionless ratio between the horizon radii
of the black hole. It is further shown that this {\it analytically} derived
upper bound on the superradiant instability regime of the spinning Kerr black
hole agrees with recent {\it numerical} computations of the instability
resonance spectrum.
