The following is taken from an old complex analysis qualifying exam.
Let $\Delta$ denote the open unit disc.
Suppose $f:\Delta\setminus\{0\}\rightarrow \mathbb{C}$ is holomorphic and assume that \int_{0<|x+iy|<1}|f(x+iy)|^2dxdy<\infty. Prove that $f$ can be extended uniquely to a holomorphic function on $\Delta$.
I would like to show that $|f(z)|$ is bounded in a neighborhood of 0, and then use Riemann's removable singularity theorem... but this is giving me trouble.
I can use Cauchy's integral formula on $f^2$ to obtain $|f(z)|^2\leq\frac{1}{\pi R^2}\int_0^{2\pi}\int_0^R|f(z+re^{i\theta})|^2r\,drd\theta,$ where R<|z|. And this double integral is no greater than the given integral, which is finite. However, the $1/R^2$ is preventing me from concluding anything about boundedness near 0.
Any help would be greatly appreciated.