1) Show by means of an example that a non-constant holomorphic function on an un-
bounded domain need not achieve it's maximum modulus on the boundary of that domain
(assuming the boundary is not empty.)
2) a) Given
where |a| 1 and ainR, compute
belong to Aut(D), the group of holomorphic automorphisms of the unit disc. Find C and
7, where k < 1 and 7 E R, such that
3) If H = € C I Im(z) > 0} is the upper half plane, use your knowledge of the el-
ements of Aut(D) to describe the elements of Aut(H) as fractional linear transformations
4) a) Show that the function
defines a homeomorphism between the unit disc D and C.
b) Show that there cannot exist any function between D and C which is a biholomorphism
(hint: Liouville's theorem).
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