We have the following result:

Let $R=\mathbb{C}[t]_f$, with $f=(t-a_1)(t-a_2)\cdots (t-a_n)$. Then the automorphism group of $R$ is isomorphic to the group of all Moebius transformations which fix (not necessary pointwise) the set $\{a_1,\ldots,a_n,\infty\}$.

Is it a known result or a direct consequence of some known theorem in algebraic geometry?