  The following inequalities are established, improving a former inequality due
to Kojima. For any closed arithmetic hyperbolic $3$--manifold fibering over a
circle, the entropy of the pseudo-Anosov monodromy is bounded by the hyperbolic
volume of the $3$--manifold, up to a universal constant factor. For any closed
hyperbolic $3$--manifold fibering over a circle with systole
$\geq\varepsilon>0$, the entropy is bounded by the hyperbolic volume times
$\log(3+1/\varepsilon)$, up to a universal constant factor. The proof relies on
Heegaard Floer homology and hyperbolic geometry.
