We study three types of the old-minimal higher-derivative supergravity theories extending the $f(R)$ gravity, towards their use for the inflationary model building in supergravity, by using both superfields and their field components. In the curved superspace all those theories are described in terms of a single chiral scalar curvature superfeld $\mathcal{R}$. Each of those theories can be dualized into a matter-coupled supergravity without higher derivatives. The first type is parametrized by a single non-holomorphic potential $N(\mathcal{R},\bar{\mathcal{R}})$, and gives rise to the dual matter-coupled supergravities with two dynamical chiral matter superfields having a no-scale K��hler potential. We find that a generic potential $N(\mathcal{R},\bar{\mathcal{R}})$ generates both the $(R+R^2)$ gravity and the non-minimal coupling of the propagating complex scalar field to the $R$, needed for the Starobinsky and Higgs inflation, respectively. We find the general conditions for the Starobinsky inflation and compute the inflaton mass. The second type is given by the chiral supergravity actions whose superfield Lagrangian $F(\mathcal{R},��({\bar{\mathcal R}}))$ also depends upon the chiral projection $��$ of the anti-chiral superfield ${\bar{\mathcal R}}$. We find that the actions of the second type always give rise to ghosts. We also revisit the $F(\mathcal{R})$ supergravity actions of the third type (without the $��$-dependence) with the reduced number of the extra physical degrees of freedom, comprising a single chiral matter superfeld with a no-scale K��hler potential. We confirm that the pure $F(\mathcal{R})$ supergravity is insufficient for realization of the Starobinsky inflation, though by the reason different from those proposed in the recent literature.

LaTeX, 24 pages, no pictures; Sections 3 and 4 expanded, references added, typos corrected