Automorphism group any bounded domain of $\mathbb{C}$

Question

So far the automorphism group I have calculated for known domain is a Lie Group,so Automorphism group any bounded domain of $\mathbb{C}$ is a lie group?

Answer 1

It is a 1935 result of Henri Cartan that the automorphism group $Aut(D)$ of a bounded domain $D\subset \mathbb C^n$ is a real Lie group.
Also, the holomorphic automorphism group of any Riemann surface is a real Lie group.
Each of these two results solves your problem but unfortunately there does not seem to be a book on holomorphic functions of one variable proving the particular case you are interested in.

In the same vein (but not directly relevant to your question), the holomorphic automorphisms of a compact manifold form a real Lie group, according to a theorem of Bochner and Montgomery.

A great specialist, W. Kaup, has written a survey article on all these results, which will also allow you to access the bibliography of the problem.

Answer 2

Every complex automorphism is an isometry in the hyperbolic metric, and, conversely, an isometry in hyperbolic metric is either holomorphic or antiholomorphic automorphism of the domain. The isometry group is a Lie group by the Myers-Steenrod theorem. Holomorphic automorphisms form a closed subgroup (of index 2), hence a Lie group of its own.