  Modified gravity models are subject to a number of consistency requirements
which restrict the form that the function $F(R)$ can take. We study a
particular class of $F(R)$ functions which satisfy various constraints that
have been found in the literature. These models have a late time accelerating
epoch, and an acceptable matter era. We calculate the Friedmann equation for
our models, and show that in order to satisfy the constraints we impose, they
must mimic General Relativity plus $\Lambda$ throughout the cosmic history,
with exponentially suppressed corrections. We also find that the free
parameters in our model must be fine tuned to obtain an acceptable late time
accelerating phase. We discuss the generality of this conclusion.
