Fixed points on Riemann surface

It is well known theorem that for a conformal mapping $\phi$ from a bounded and planar domain $\Omega$ to itself has three fixed points , then it must be identity mapping. However, I cannot find a elementary proof for it?

Any comments and reference will be appreciated

This theorem is an immediate consequence of a result by B. Maskit, which states that one may associate with $\Omega$ another domain $\Omega'$, conformally equivalent to $\Omega$, such that all conformal self-maps of $\Omega'$ are Möbius transformations. For a proof of Maskit's theorem, I suggest you look at the following papers :