Let $u(x)$ be a harmonic polynomial in the unit ball $B_1(0)\subset\mathbb{R}^n$ with $u(0)=0$. 

For $0<r\leq1$, consider the average of its Dirichlet integral 
$$A(r):=\frac1{\vert B_r(0)\vert}\int_{B_r(0)}\vert\nabla u\vert^2dx,$$
and the average of the square function on the boundary
$$B(r):=\frac1{\vert \partial B_r(0)\vert}\int_{\partial B_r(0)}u^2d\sigma.$$

I would like to ask:

>**QUESTION.** Is this true? The ratio $\frac{r^2A(r)}{B(r)}$ is a constant in $r$.

**POSTSCRIPT.** Iosif Pinelis responded properly (thank you!). On the other hand, I realized what I missed from the assumption: $u(x)$ should be a **homogeneous** harmonic polynomial (thinking of spherical harmonics). Sorry about that. I am hoping someone can try addressing that here, instead of initiating a new MO post.