bf MATH 6322, Complex Analysis Fall 2011, Key
to HW#6
October 31, 2011
Additional Problems.
1. Let
a
∈
D
(0
,
1), and define the M¨
obius transform:
φ
a
(
z
) =
z

a
1

¯
az
Prove that
a).
φ
a
maps the unit disc into unit disc, and
φ
a
is holomorphic on
D
(0
,
1).
b). Prove that
φ
a
:
D
(0
,
1)
→
(0
,
1) is onetoone and onto and its inverse is
φ

a
.
c). Use a), b) and Schwartz Lemma to prove the SchwartzPick theorem.
Proof.
a). It is clear that
φ
a
(
z
) =
z

a
1

¯
az
is holomorphic on
D
(0
,
1), since 1

¯
az
has no
zeros on
D
(0
,
1).
To show that
φ
a
maps the unit disc into unit disc, for
z
∈
D
(0
,
1), we compare

z

a

2
with

1

¯
az

2
, and have

z

a

2
 
1

¯
az

2
=

z

2

2
<
¯
az
+

a

2

(1

2
<
¯
az
+

a

2

z

2
)
=

z

2
+

a

2

1
 
a

2

z

2
= (1
 
z

2
)(

a

2

1)
<
0
since

z

<
1 and

a

<
1. Hence,

z

a

2
 
1

¯
az

2
, which implies
z

a
1

¯
az
<
1,
i.e.

φ
a
(
z
)

<
1
,
for each
z
∈
D
(0
,
1)
.
b). For each
w
∈
D
(0
,
1), we solve
φ
a
(
z
) =
w
, i.e.
z

a
1

¯
az
=
w,
1
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to get
z
=
w
+
a
1 + ¯
aw
.
Using (
a
) above, we have that

z

<
1. Since
z
is unique for each
w
∈
D
(0
,
1),
φ
a
is onetoone and onto, and the inverse of
φ
a
(
z
) is
φ

a
(
w
) =
w
+
a
1 + ¯
aw
,
w
∈
D
(0
,
1)
.
c). Assume
f
(
a
1
) =
b
1
and
f
(
a
2
) =
b
2
, then we take
h
(
z
) =
φ
b
1
◦
f
◦
φ

a
1
(
z
)
So
h
is a holomorphic function with
h
(0) = 0 and

h
(
z
)

<
1 for all
z
∈
D
(0
,
1),
and using the Schwartz Lemma gives us

h
(
z
)

=

φ
b
1
◦
f
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 Addition, Power Series, Taylor Series, Pole, z

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