[en] The measurements recently performed by the Viking spacecraft have shown that, in addition to being cold plasma depleted, the source regions of the Auroral Kilometric Radiation (A.K.R.) are characterized by a relatively denser, more energetic electron component. In order to properly study the Cyclotron Maser Instability (C.M.I.) which is thought to be responsible for the A.K.R. generation, it is thus necessary to include relativistic corrections in both the hermitian and the antihermitian parts of the dielectric tensor characterizing the linear properties of the plasma. Here one presents an analytical study of the corresponding dispersion equation which aims to describe stable and unstable waves having frequencies lying very close to the electronic gyrofrequency and propagating across the geomagnetic field with a perpendicular refractive index less than a few units (n_{perpendicular} < 5). The electronic population of the auroral plasma is supposed to have two components. The distribution function of the non-thermal, energetic component is modelled by a Dory-Guest-Harris (DGH) distribution function and those of the tenuous thermal one by either a Dirac's function or a maxwellian. The calculations allow the authors to simply perform a parametric study of the various regimes presented by the CMI. For the range of parameters inferred from measurements of the Swedish spacecraft Viking: (P > 1 and χ small), the growth rate could maximize at the cut-off frequency of the relativistic X mode. Moreover, for small χ, the relativistic X mode is connected to freely propagating modes which guarantees an easy access of the electromagnetic energy to free space