Convolution comparison measures

We give a precise functional comparison between classical and free convolutions. If $\mu$ and $\nu$ are compactly supported probability measures, we show that the expectation of $f$ over the classical convolution $\mu * \nu$ is at least the expectation of $f$ over the free convolution $\mu \boxplus \nu$, as long as the fourth derivative of $f$ is non-negative. Conversely, the non-negativity of the fourth derivative is necessary for such a comparison. This comparison is based on the positivity of a related measure on $\mathbb{R}^{2}$, which we dub the convolution comparison measure. We give an expression for this measure using a curious identity involving Hermitian matrices.