Key to Complex Analysis Homework 3
September 27, 2011
Chapter 1 (in Greene & Krantz’s book)
29. Suppose that
f
:
D
(0
,
1)
→
C
is holomorphic and that

f
 ≤
2. Derive an
estimate for
∂
3
∂z
3
f
i
3
.
Proof.
Choose 0
< r <
2
/
3 so that
D
(
i
3
, r
)
⊂
D
(0
,
1).
By Cauchy’s esti
mate,
∂
3
∂z
3
f
i
3
≤
2
*
3!
r
3
.
Letting
r
→
2
/
3, we get
∂
3
∂z
3
f
i
3
≤
2
*
3!
(
2
3
)
3
=
81
2
.
2
32. Suppose that
f
is bounded and holomorphic on
C
\ {
0
}
. Prove that
f
is
constant. [
Hint:
Consider the function
g
(
z
) =
z
2
·
f
(
z
) and endeavor to apply
Theorem 3.4.4]
Proof.
We consider the following auxiliary function,
g
(
z
) =
z
2
f
(
z
)
,
if
z
6
= 0;
0
,
if
z
= 0
.
Since
f
is holomorphic on
C
\{
0
}
, so is
g
on
C
\{
0
}
. However, at
z
= 0, we need
to check whether
g
0
(0) exists. By the condition that
f
is bounded on
C
\ {
0
}
,
we obtain
g
0
(0) = lim
z
→
0
g
(
z
)

g
(0)
z

0
= lim
z
→
0
z
2
f
(
z
)
z
= lim
z
→
0
zf
(
z
) = 0
.
1
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Hence, we know that
g
is holomorphic on entire
C
.
Choose
M
such that

f
 ≤
M
, on
C
\ {
0
}
, we have

g
(
z
)
 ≤
M

z

2
,
for all
z
∈
C
this implies, by Theorem 3.4.4,
g
is a complex polynomial of at most 2 degree.
Write as
z
2
f
(
z
) =
g
(
z
) =
a
0
+
a
1
z
+
a
2
z
2
on
C
\ {
0
}
. Let letting
z
→
0 and
noticing that
f
is bounded, we get
a
0
= 0. Hence
z
2
f
(
z
) =
a
1
z
+
a
2
z
2
on
C
\{
0
}
or
zf
(
z
) =
a
1
+
a
2
z
. Again, by letting
z
→
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 Summer '08
 Staff
 Fundamental Theorem Of Algebra, Complex number, Compact space, Uniform convergence, Entire function

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