Hello. I am looking at the following problem:

Let $\displaystyle f(z)$ be holomorphic in $\displaystyle D(0,1)/\{0\}$ so that

$\displaystyle \displaystyle \int_0^1 |f(re^{i\theta})|^2\text{ d}\theta \leq 1, \text{ for all } 0<r<1$

Prove that $\displaystyle z=0$ is a removable singularity.

I'd like to use the Gauss Mean Value Property, where

$\displaystyle \displaystyle f(0)=\frac{1}{2\pi} \int_0^{2\pi} f(re^{i\theta})\text{ d}z$

Then I could try to put a bound on $\displaystyle f$ and conclude that the singularity is removable. However, the Gauss Mean Value property only works if the function is holomorphic on a simply connected domain.

I was wondering if there are any suggestions on how to approach this problem. Any ideas would be much appreciated.