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In this paper, we find quasinormal spectrum of the massive scalar field in the background of the Kerr black holes. We show that all found modes are damped under the quasinormal modes boundary conditions when 1, thereby implying stability of the massive scalar field. This complements the region of stability determined by the Beyer inequality for large masses of the field. We show that, similar to the case of a nonrotating black hole, the massive term of the scalar field does not contribute in the regime of high damping. Therefore, the high damping asymptotic should be the same as for the massless scalar field.
Konoplya et al. (Thu,) studied this question.