The L mode in electromagnetic proton-cyclotron waves (EPCWs) propagating parallel to a uniform ambient magnetic field is studied here analytically. A generalized Lorentzian distribution function is used to model the plasma. Analytical expressions for the wavenumber and for both the temporal and convective growth rates for a multi-ion plasma are obtained within the linear theory. This analytical approach is appropiate for β∥<1, which is the ratio of plasma kinetic pressure to magnetic field pressure. The characteristics of the unstable spectrum are found to be independent of high-energy particles. For a plasma composed of electrons plus hot and cold protons, it is shown that the maximum growth rates as functions of cold-proton concentration δ can always decrease, or can increase until δ reaches a certain peak value and decrease thereafter, or can always increase, depending on the thermal anisotropy of the hot protons. This behaviour is similar to that in Maxwellian plasmas. However, for the convective growth rate, the expression for the optimum cold-proton concentration shows a significant dependence on the spectral index κ. Therefore, when cold protons are injected, it is more difficult to obtain optimum amplification in a Lorentzian plasma than in a Maxwellian plasma. It is also shown that the influence of the high-energy tail on the generation and amplification processes of the EPCWs is controlled by thermal anisotropy and cold-ion population. As a consequence of the latter, temporal and convective growth rates can be larger than, equal to or smaller than those of Maxwellian plasmas, depending on the anisotropy of the hot-proton distribution and on the cold-proton concentration.