Inequality for Voiculescu's free entropy in terms of Brown measure

We study Voiculescu’s microstate free entropy for a single non-selfadjoint random variable. The main result is that certain additional constraints on eigenvalues of microstates do not change the free entropy. Our tool is the method of random regularization of Brown measure which was studied recently by Haagerup and the author. As a simple application we present an upper bound for the free entropy of a single non–selfadjoint operator in terms of its Brown measure and the second moment. We furthermore show that this inequality becomes an equality for a class of DT-operators which was introduced recently by Dykema and Haagerup.