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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 a = 1-az and po(2)=ee..2 where |a| 1 and ainR, compute (6.0p.-pood)1(2) b) Suppose f(2)=00po(2)and 9(2)=dropp(z) 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 fog=00 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 z = 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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