  This work is devoted to direct mass transportation proofs of families of
functional inequalities in the context of one-dimensional free probability,
avoiding random matrix approximation. The inequalities include the free form of
the transportation, Log-Sobolev, HWI interpolation and Brunn-Minkowski
inequalities for strictly convex potentials. Sharp constants and some extended
versions are put forward. The paper also addresses two versions of free
Poincar\'e inequalities and their interpretation in terms of spectral
properties of Jacobi operators. The last part establishes the corresponding
inequalities for measures on $\R_{+}$ with the reference example of the
Marcenko-Pastur distribution.

Functional Analysis;Probability