Three spheres inequality

In mathematics, the three spheres inequality bounds the $$L^2$$ norm of a harmonic function on a given sphere in terms of the $$L^2$$ norm of this function on two spheres, one with bigger radius and one with smaller radius.

Statement of the three spheres inequality
Let $$u$$ be an harmonic function on $$\mathbb R^n$$. Then for all $$0 < r_1 < r 0$$ is the sphere of radius $$\rho$$ centred at the origin and where
 * $$\alpha:=\frac{\log(r_2/r)}{\log(r_2/r_1)}.$$

Here we use the following normalisation for the $$L^2$$ norm:
 * $$ \| u \|^2_{L^2(S_\rho)} := \rho^{1-n} \int_{\mathbb S^{n-1}} \vert u(\rho \hat x) \vert^2\, d\sigma(\hat x).$$