 Uniform L2 circle bound for a removable singularity

If $f$ is holomorphic on $0<|z|<R$ andthen the singularity at zero is removable. The Laurent coefficient formula and Cauchy--Schwarz give $|a_{-k}|=O(r^k)$ for each $k\ge 1$, so every principal-part coefficient vanishes as $r\downarrow 0$.