Given a lower bound on the Ricci curvature of a complete Riemannian manifold $M$, the Bishop-Gromov inequality allows us to deduce an upper bound on the volume of (geodesic) balls in $M$.
Is there a similar statement if we assume an upper bound on the Ricci curvature of $M$? I.e., does such an upper bound allow us to deduce a lower bound on the volume of balls in $M$?
If it is easier, feel free to restrict your attention to just Hadamard manifolds.