  We revisit the problem of generality of Starobinsky and Higgs inflation. The
known results obtained in the Einstein frame are generalized for the case of an
arbitrary initial energy of the scalar field. These results are compared with
the results obtained directly in the Jordan frame, which, to our knowledge, has
not been thoroughly explored in the literature previously. We demonstrate that
the qualitative picture of initial conditions zone in the $(\phi, \dot \phi)$
plane, which leads to sufficient amount of inflation, is quite similar for both
the frames in the case of Higgs inflation. For Starobinsky inflation, the
conformal transformation between the frames relates the geometrical variables
in the Jordan frame with the properties of an effective scalar field in the
Einstein frame. We show that the transformation $(H, R) \to (\phi, \dot \phi)$
is not regular everywhere, leading to some peculiarities in the zone of good
initial conditions in the $(H, R)$ plane.

General Relativity and Quantum Cosmology;Cosmology and Nongalactic Astrophysics;High Energy Physics - Phenomenology;High Energy Physics - Theory