MATH 5321
Exam II
1 April 2009
Answer the problems on separate paper. You do not need to rewrite the problem statements on
your answer sheets. Work carefully. Do your own work. Show all relevant supporting steps!
1. (30 pts)
State and prove Rouché’s Theorem.
2. (30 pts)
Give the definition for each of the following:
a.
Let f , g ∈ A ( D ) . Then, f is subordinate to g ( f ≺ g ) if . . .
b.
For { K n } a compact exhaustion of a region G and for f , g ∈ C (G , Ω ) let
ρ( f , g) = . . .
c.
Let G be a region. A function f is meromorphic on G if . . .
d.
A set F ⊂ C (G , Ω ) is normal . . .
e.
Let G be a region and let f : G →
Property (MVP) on G if . . .
f.
A set F ⊂ C (G , Ω ) is equicontinuous at a point z 0 ∈ G if . . .
3. (20 pts)
Find the number of zeros of p ( z ) =
4. (20 pts)
Let G be a bounded region in
11
10
. f satisfies the Mean Value
z 6 − 265 z 5 + 3 z 2 − 1 in | z | ≤ 1 .
. Let { f n } ⊂ C (G ,
) ∩ A (G ) and let
f ∈C (G , ) ∩ A (G ) . Suppose that f n → f uniformly on ∂G . Show
that f n → f in C (G ,
).