  We establish the mean-field bounds $\gamma \ge 1$, $\delta \ge 2$ and
$\triangle \ge 2$ on the critical exponents of the Poisson-Boolean continuum
percolation model under a moment condition on the radii; these were previously
known only in the special case of fixed radii (in the case of $\gamma$), or not
at all (in the case of $\delta$ and $\triangle$). We deduce these as
consequences of the mean-field bound $\beta \le 1$, recently established by
Duminil-Copin, Raoufi and Tassion under the same moment condition, using a
relative entropy method introduced by the authors in previous work.
