TITLE: Singularities in the punctured unit disc and square integrability
QUESTION [6 upvotes]: I'm having trouble with the following problem:
Let $f$ be holomorphic on the punctured unit disc, $D$. If $\int_D|f(z)|dA(z)<\infty$, then $z=0$ is either a removable singularity or a simple pole of $f$.
A similar problem is to prove or disprove that if $\int_D|f(z)|^2dA(z)<\infty$, then $f$ has a removable singularity at $0$.
I've tried using the Mean Value Property for $f$ and taking the limit as $z\to0$, but I didn't get anywhere.

REPLY [2 votes]: For the first problem, write $c_n$ as an integral around the circle of radius $r$ centred at $0$.  If $\int_D |f(z)|\ dA < \infty$, the integral of that times a certain power of $r$ for $r$ from 0 to 1 would be finite ...