An elementary proof of the inequality ^* for conditional free entropy

Through the study of large deviations theory for matrix Brownian motion, Biane, Capitaine, and Guionnet proved the inequality (X) ^* (X) that relates two analogs of entropy in free probability defined by Voiculescu. We give a new proof of ^* that is elementary in the sense that it does not rely on stochastic differential equations and large deviations theory. Moreover, we generalize the result to conditional microstates and non-microstates free entropy.