The automorphism group of the punctured disc $A=\mathbb{D}-\{a\}$ consists of all holomorphic maps $f$ from $A$ onto itself, and since $A$ is bounded, then $f$ should be bounded, and so $a$ is removable singularity and hence $f$ extends to the automorphism on $\mathbb{D}$ .
Can I extend this argument to the punctured disc $B=\mathbb{D}-\{a,b\}$? with two isolated singularities $a, b$.