  In this paper we deal with free functional inequalities on the circle. There
are some interesting changes as opposed to the classical case. For example, the
free Poincar\'e inequality has a slight change which seems to account for the
lack of invariance under rotations of the base measure. Another instance is the
modified Wasserstein distance on the circle which provides the tools for
analyzing transportation, Log-Sobolev, and HWI inequalities.
  These new phenomena also indicate that they have a classical counterpart,
which does not seem to have been investigated before.
