  We study the concavity of the dispersion relation $q\mapsto
\omega\_{\mathbf{q}}$ of the bosonic excitations of a three-dimensional
spin-$1/2$ Fermi gas in the Random Phase Approximation (RPA). In the limit of
small wave numbers $q$ we obtain analytically the spectrum up to order $5$ in
$q$. In the neighborhood of $q=0$, a change in concavity between the convex BEC
limit and the concave BCS limit takes place at $\Delta/\mu\simeq0.869$
[$1/(k\_F a)\simeq-0.144$], where $a$ is the scattering length between opposite
spin fermions, $k\_F$ is the Fermi wave number and $\Delta$ the gap according
to BCS theory, and $\mu$ is the chemical potential. At that point the branch is
concave due to a negative fifth-order term. Our results are supplemented by a
numerical study which shows the evolution of the border between the zone of the
$(q,\Delta)$ plane where $q\mapsto \omega\_{\mathbf{q}}$ is concave and the
zone where it is convex.
