Is this Laplacian comparison? Cheeger-Colding-Minicozzi: 1995 Linear growth harmonic functions on complete manifolds with nonnegative ricci curvature in GAFA
Page 952:
From Laplacian comparison, we have for $r<R$,
\begin{equation*}
\begin{aligned}
0 &=\int_{B_r(p)}\Delta h_R=\int_{\partial B_r(p)}\frac{\partial h_R}{\partial r}\\
&\ge \frac{\partial}{\partial r}\int_{\partial B_r(p)} h_R -\frac{n-1}{r}\int_{\partial B_r(p)} h_R
\end{aligned}
\end{equation*}
where $h_R$ is a harmonic function on $B_R(p)\subset M^n$, $M^n$ is open Riemannian manifold with nonnegative Ricci curvature, and $0<r<R$.
I don't know how to derive the inequality.

If $J(r,\theta) dr \wedge d\theta$ is volume form then $\frac{\partial}{\partial r}J(r,\theta) = H J$ where $H$ is a mean curvature. 
 If $M$ has a nonnegative sectional curvature, then $H \leq \frac{n-1}{r}$.
Whence $\frac{\partial}{\partial r}\int_{\partial B_r(P)} h_R \leq \int_{\partial B_p(r)} \frac{\partial}{\partial r} h_R  +  \int_{\partial B_r(p)} h_R 
\frac{\partial}{\partial r} J dr\wedge d\theta$
$ \leq   
 \int_{\partial B_p(r)} \frac{\partial}{\partial r} h_R  +  \int_{\partial B_r(p)} h_R H J dr\wedge d\theta $
$\leq   \int_{\partial B_p(r)} \frac{\partial}{\partial r} h_R  +  \frac{n-1}{r} \int_{\partial B_r(p)} h_R $
And $ \int_{ B_p(r)} \Delta  h_R =   \int_{\partial B_r(p)} 
\frac{\partial}{\partial r} h_R $ is a divergence theorem.   
