Complex Analysis : Analytic Functions as Mappings

1). Let G be a region and suppose that f : G -> C ( C is complex plane) is analytic such that f(G) is a subset of a circle. Show that f is constant.

2). If Tz = (az + b)/(cz + d), find necessary and sufficient conditions that T(t) = t where t is the unit circle { z: |z| = 1}.
My solution for number 2 is :
T(t) = t , which implies that | (az+b/cz+d| = 1 then we have |az+b| = |cz+d| then by solving for z we get |d-b|= |a-c| or |d+b| = |a+c|. Am I right? If not, please provide the correct solution.
( I believe z here is a complex number)

Please work these problems only if you are an expert complex math person.