This inequality (as it is formulated) is not true in the general case. Here is an example in two dimensions. Your notation $\overline{u}(r,w)$ is not proper because  the average of $u(r,w)$ over the sphere does not depend on $w$. Let us define $$u(r,\phi):=\begin{cases}
1,\mbox{ for }\phi \ge 0 \mbox{ and }\phi \le \epsilon,\\
0, \mbox{ otherwise.}\\
\end{cases}$$ Then $\overline{u}(r)= \frac \epsilon {2\pi}.$ Therefore, the best possible constant $C$ is $\frac {2\pi} \epsilon$, and depends on $u$. Such inequality may be true under additional assumptions on $u$. For instance, it is true with the constant $C=1$ for [subharmonic functions](http://en.wikipedia.org/wiki/Subharmonic_function).