Cyclotron wave-particle interactions in the case of well-organized distributions of energetic electrons under an inhomogeneous magnetic field are studied. Step-like and Δ function distributions over the field-aligned velocity are considered. The one-hop amplification of whistler waves is calculated by simple analytical solution and numerical computation based on strict approach. The strict consideration, taking into account third-order expansion of the spatial dependence of the electron phase with respect to the wave, reveals some new important features of the one-hop amplification &Ggr; as a function of frequency and electron beam parameters. The main result is that &Ggr; exhibits a quasi-periodic structure as a function of wave frequency or characteristic electron parallel velocity, remaining always positive in the case of the step-like distribution but being sign alternative for Δ-function. Dependence of &Ggr; on the parameters of energetic electrons such as their total energy, characteristic parallel velocity, position of the injection point in relation to the equator, and dispersion in parallel velocity is discussed. ¿ 1998 American Geophysical Union