Dispersion relations are calculated and solved for frequencies near the first two harmonics (Ω/Ω+ = 2, 3) of perpendicular magnetosonic waves in relativistic anisotropic plasmas. In this, wavelengths larger than the ion Larmor radius and large plasma frequency (Ω2p, ≫ Ω2+) are assumed. It is found that, even for Tψ ≫ T the plasma is stable. This result has application to the steady-state, spherically symmetrical, expanding, solar-wind plasma.