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 $r$.
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> 
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A constant ratio of integrals? Part I
No. E.g., if $n=2$ and $u(x,y)=x + x^2 - y^2$ for $(x,y)\in\mathbb R^2$, then $\dfrac{r^2A(r)}{B(r)}=2\dfrac{1+2r^2}{1+r^2}$.


