  Economical extensions of the Standard Model (SM), in which the famous
Davidson-Ibarra bound on the CP asymmetry relevant for leptogenesis may be
significantly relaxed by the loop effects, comparing to predictions of the SM
extended only by right-handed neutrinos with hierarchical masses, are
discussed. This leads to decreasing of the lower bound on the heavy neutrino
masses and increasing of the upper bound on the light neutrino masses. In
addition, the considered theory may help to solve the dark matter problem.
