In this work we construct a formalism that can reveal the general characteristics of classes of viable F (R) inflationary theories. The assumptions we make is that the slow-roll era occurs, and that the de Sitter scalaron mass m² (R) of the F (R) gravity is positive or zero, for both the inflationary and late-time quasi de Sitter eras, a necessary condition for the stability of the de Sitter spacetime. In addition, we require that the de Sitter scalaron mass is also a monotonically increasing function of the Ricci scalar, or it has an extremum. Also the F (R) gravity function is required to depend on the two known fundamental scales in cosmology, the cosmological constant Λ and the mass scale mₛ²=κ² ρₘ^ (0) 3, with ρₘ^ (0) denoting the energy density of the cold dark matter at the present epoch, that is F (R) =F (R, Λ, mₛ²). Using these general assumptions we provide the general features of viable classes of F (R) gravity inflationary theories which remarkably can also simultaneously describe successfully the dark energy era. This unique feature of a unified description of the dark energy and inflationary eras stems from the requirement of the monotonicity of the de Sitter scalaron mass m² (R). These viable classes are either deformations of the R² model or α-attractors type theories. The analysis of the viability of a general F (R) gravity inflationary theory is reduced in evaluating the parameter x=R FₑₑₑFₑₑ and the first slow-roll index of the theory, either numerically or approximately. We also disentangle the power-law F (R) gravities from power-law evolution.
V. K. Oikonomou (Tue,) studied this question.