Harmonic function squared and mean value

[arestes]

Homework Statement

Let u be a harmonic function in the open disk K centered at the origin with radius a. and
[itex]∫_K[u(x,y)]^2 dxdy = M < ∞[/itex]. Prove that
[itex]|u(x,y)| \le \frac{1}{a-\sqrt{x^2+y^2}}\left( \frac{M}{\pi}\right)^{1/2}[/itex] for all (x,y) in K.

Homework Equations

Mean value property for harmonic functions.

The Attempt at a Solution

I first thought this was easy and directly applied the mean property for harmonic functions, but of course, the square of a harmonic function is not harmonic (unless it's a constant).

I can see this problems begs for the mean value using a ball with radius [itex]a-\sqrt{x^2+y^2}[/itex] which would be entirely inside the disk K but I can't get around the fact that u squared is not harmonic. Help please

 

[brmath]

Are you integrating over the whole disk K (i.e. a double integral) or as a line integral around the boundary of K.

 

[arestes]

I integrated over the area of the smaller circle inside with radius [itex] r=a-\sqrt{x^2+y^2}[/itex]. This is exactly what I did (I know it's wrong):

[itex] u^2(x,y)=\frac{\int_{B_r(x,y)} u^2(w, z)dwdz}{\pi r^2}\leq \frac{\int_K u^2(w, z)dwdz}{\pi r^2}=\frac{M}{\pi r^2}\\
|u(x,y)|\leq \sqrt{\frac{M}{\pi r^2}}[/itex]
But of course, this would work if u^2 were harmonic, which is seldom the case.

Any ideas? I was thinking maybe applying the maximum principle and somehow get bounds...

 

[brmath]

I think the answer is lurking here:

http://unapologetic.wordpress.com/2009/12/29/the-mean-value-theorem-for-multiple-integrals/

This is the mean value theorem for multiple integrals and provides some bounds on the integrals. The functions do not have to be harmonic. I think if you start here you will eventually reach a place where u being harmonic will refine the inequalities.

Worth a shot.