Please help me to prove.
Prove that if domain conformaly maps to unit disc then it's simply connected domain.
Thanks!
You need to assume that the conformal map is bijective (and therefore invertible), otherwise the result is not true.
Suppose that S is a loop in . The image f(S) of S is a loop in the simply-connected space D. So there is a continuous transformation in D that contracts this loop to a point. The image in of that deformation under the map will be a continuous transformation in which contracts S to a point S to a point.