Hints to Complex Analysis Homework 5
October 3, 2011
8.
You want to compare the zeros of
f
and
g
inside
D
(
P,r
), so, you want

f

g

<

f

on
∂D
(
P,r
), obviously,

f

g

<
min
z
∈
∂D
(
P,r
)

f
(
z
)

will do it
(here since
f
has no zeros on
∂D
(
P,r
), min
z
∈
∂D
(
P,r
)

f
(
z
)

>
0). So here the
meaning
g
is uniformly suﬀciently close to
f
on
∂D
(
P,r
) means we require
that, on
∂D
(
P,r
),

f

g

<
min
z
∈
∂D
(
P,r
)

f
(
z
)

.
10
(b)
f
(
z
) =
z
3

3
z
2
+ 2 on
D
(0
,
1).
Solution
Method 1 (not very pretty)
: Observe that
z
= 1 is a solution, so we can
write
f
(
z
) = (
z

1)(
z
2

2
z

2), and we only need to consider the zero of
z
2

2
z

2 on
D
(0
,
1). by the quadaritic formula,
z
2

2
z

2 = 0 has two roots
1 +
√
3 and 1

√
3, and 1

√
3 is inside
D
(0
,
1). So
f
has only one zero in
D
(0
,
1)
Method 2 (Use Rouche’s theorem)
: Note that
f
has at most three zeros
on
D
(0
,
1), so we can take