Three spheres inequality

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

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Statement of the three spheres inequality

Let be an harmonic function on . Then for all one has

where for is the sphere of radius centred at the origin and where

Here we use the following normalisation for the norm:

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