Picture bottom is from Topping's Lectures on the Ricci flow. I can't get the red line.
First, I have the Bishop's theorem:
where $V^\alpha(r)$ is the volume of a ball of radius $r$ in the complete simply connected Riemannian manifold with constant curvature $\alpha$.
For the red line, I know the sectional curvature $K\ge -1$. But only use the (i) of Bishop-Gunther theorem, I can get the above boundary of $\frac{\nu(p,r)}{r^n}$. Therefore, how to get the low boundary of $Ric$ ?
Since $Ric(x)=\frac{1}{n-1}\sum \langle R(x, z_i)x, z_i\rangle $ (according to do Carmo's Riemannian Geometry), I have $Ric \ge -g $. Consider the irregular use of notations, I think the $a$ of (i) of Bishop-Gunther is $-1$. Therefore, I have $$ \nu(p,r)=Vol(B_M(r))\le V^{-1}(r) $$ So, I can get the above boundary of $\frac{\nu(p,r)}{r^n}$. I am not sure whether it is right, so ask here.
Besides, how to calculate $V^{-1}(r)$ ? Although I guess that it is $Cr^n$, but I don't know how to calculate $V^{-1}(r)$ ?
