Upper bound on volume growth

Question

If the Ricci curvature of a compact Riemannian manifold of demsnion $n$ is greater than 1-n, does it follow that the volume entropy satisfies $$\liminf_{r\rightarrow \infty} \frac{\log vol B_r(p)}{r}\leq n-1$$

Answer 1

Yes, this follows from the Bishop-Gromov volume comparison theorem.