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