  We generalize the well-known analogies between m^2 \phi^2 and R + R^2
theories to include the selfinteraction \lambda \phi^4-term for the scalar
field. It turns out to be the R + R^3 Lagrangian which gives an appropriate
model for it. Considering a spatially flat Friedman cosmological model, common
and different properties of these models are discussed, e.g., by linearizing
around a ground state the masses of the resp. spin 0-parts coincide. Finally,
we prove a general conformal equivalence theorem between a Lagrangian L = L(R),
L'L" \ne 0, and a minimally coupled scalar field in a general potential.
