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

Mass Transportation Proofs of Free Functional Inequalities, and Free Poincare Inequalities