  We investigate the cosmological inflation in a class of supergravity models
that are generalizations of non-supersymmetric $R^2$ models. Although such
models have been extensively studied recently, especially after the launch of
the PLANCK and BICEP2 data, the class of models that can be constructed has not
been exhausted. In this note, working in a supergravity model that is a
generalization of Cecotti's model, we show that the appearance of new
superpotential terms, which are quadratic in the superfield $\, \Lambda$ that
couples to the Ricci supermultiplet, alters substantially the form of the
scalar potential. The arising potential has the form of the Starobinsky
potential times a factor that is exponential in the inflaton field and
dominates for large inflaton values. We show that the well-known Starobinsky
inflation scenario is maintained only for unnaturally small fine-tuned values
of the coupling describing the $\Lambda^2$ superpotential terms. A welcome
feature is the possible increase of the tensor to scalar ratio $r$, within the
limits set by the new Planck and BICEP2 data.
