A general method is presented for deriving complex eigenvalue bounds for linear second order systems. It is applied to the problem of stratified horizontal magnetohydrodynamic shear flow, and it is fuund that. if the system is convectively stable, the imaginary part of the eigenfrequency is bounded by where uo is the flow velocity. In the absence of a magnetic field. the classical Miles-Howard-Chimonas result is recovered. but with more detail about the growth rates of individual modes. These results are shown to hold for free as well as rigid boundaries.