  We investigate the corrections to the inflationary cosmological dynamics due
to a $R^2$ term in the Palatini formulation which may arise as quantum
corrections to the effective Lagrangian in early universe. We found that the
standard Friedmann equation will not be changed when the scalar field is in the
potential energy dominated era. However, in the kinetic energy dominated era,
the standard Friedmann equation will be modified and in the case of closed and
flat universe, the Modified Friedmann equation will automatically require that
the initial kinetic energy density of the scalar field must be in sub-Planckian
scale.
