Free analogues of the logarithmic Sobolev inequality compare the relative free Fisher information with the relative free entropy. In the present paper such an inequality is obtained for measures on the circle. The method is based on a random matrix approximation procedure, and a large deviation result concerning the eigenvalue distribution of special unitary matrices is applied and discussed.

We prove a self-adjoint analogue of the Marcus--Pisier inequality, comparing the expected value of convex functionals on randomreflection matrices and on elements of the Gaussian orthogonal (or unitary) ensemble.