The fundamental resonances of near-extremal Kerr black holes due to massive
scalar perturbations are derived {\it analytically}. We show that there exists
a critical mass parameter, μc, below which increasing the mass μ of
the field increases the oscillation frequency ℜ(ω) of the resonance.
On the other hand, above the critical field mass increasing the mass μ
increases the damping rate ℑ(ω) of the mode. We confirm our analytical
results by numerical computations.Comment: 6 page