  We study several extensions of the Starobinsky model of inflation, which obey
all observational constraints on the inflationary parameters, by demanding that
both the inflaton scalar potential in the Einstein frame and the $F(R)$ gravity
function in the Jordan frame have the explicit dependence upon fields {\it and}
parameters in terms of elementary functions. Our models are continuously
connected to the original Starobinsky model via changing the parameters. We
modify the Starobinsky $(R+R^2)$ model by adding an $R^3$-term, an $R^4$-term,
and an $R^{3/2}$-term, respectively, and calculate the scalar potentials, the
inflationary observables and the allowed limits on the deformation parameters
by using the latest observational bounds. We find that the tensor-to-scalar
ratio in the Starobinsky model modified by the $R^{3/2}$-term significantly
increases with raising the parameter in front of that term. On the other side,
we deform the scalar potential of the Starobinsky model in the Einstein frame
in powers of $y=\exp\left(-\sqrt{\frac{2}{3}}\phi/M_{Pl}\right)$, where $\phi$
is the canonical inflaton (scalaron) field, calculate the corresponding $F(R)$
gravity functions in the two new cases, and find the restrictions on the
deformation parameters in the lowest orders with respect to the variable $y$
that is physically small during slow-roll inflation.
