Complex open mapping & conformal mapping problems.

1. Let f:D(0,2) -> C be a holomorphic function. Suppose that f is real-valued
on the set {|z| = 1}. Show that f is a constant.
2. Can you find a 1-1 conformal map from {z complex number | 0 < |z| < 1}
onto the annulus X = {z complex | 1 < |z| < 2}.

Thanks for helping.

Domain of a function in 1. is a disc centered at 0 and radius 2. ^^'