MATH 621 – Spring 2005
Homework Set # 10
81) Exercise 4, page 248.
82) Exercise 5, page 248.
83) Exercise 8, page 249.
84) Exercise 9, page 249.
85) Exercise 10, page 249.
86) Exercise 12, page 250.
87) Exercise 13a), page 251.
88) Exercise 16, page 252.
89) Let Ube a connected, simply-connected open subset U⊂C. Show
that given any two points Pand Qin Uthere exists an automorphism f
of Usuch that f(P) = Q. In other words, the group of automorphisms
of Uacts transitively on U.
90) Let Ω be a bounded connected open subset of Cand let Fbe
a family of functions holomorphic on Ω. Suppose that there exists
M∈Rsuch that Z ZΩ
|f(x+iy)|dxdy ≤M
for all f∈ F. Prove that Fis a normal family.
Hint: Use Problem 34 to show that Fis uniformly bounded in compact
subsets of Ω and apply Montel’s theorem.
Note: Students taking the manifolds course will find Problem 3 in
page 256 very interesting.