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MATH 6322, Complex Analysis Fall 2011, HW#6
Due date: Monday, Oct 24, 2011
Extra Problem
:
Extra 1
: Let
a
∈
D
(0
,
1) (i.e.

a

<
1) and deﬁne (such map is called the
Mobius transformation)
φ
a
(
z
) =
z

a
1

¯
az
.
Prove
(a)
φ
a
maps the unitdisc to the unit disc, i.e.

φ
a
(
z
)

<
1 for all
z
∈
D
(0
,
1). Also prove that
φ
a
is holomorphic on
D
(0
,
1).
(b) Prove that
φ
a
:
D
(0
,
1)
→
D
(0
,
1) is onetoone and onto and its
inverse is
φ

a
.
(c) Use (a) and (b) and the Schwarz lemma (Proposition 5.5.1 on P. 171)
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Unformatted text preview: to prove the SchwarzPick Theorem (Theorem 2.5.2 on Page 172). Extra 2 (Hint: Use SchwarzPick Theorem) : Let f be holomorphic on D (0 , 1) ⊂ U and  f ( z )  ≤ 1 on D (0 , 1). Prove that, for all z ∈ D (0 , 1), (1  z  2 )  f ( z )  ≤ 1 . Chapter 4: 5 (a), (b), (e), 13 (a), (b), (f), 27 (a), (b), (c), (h). 1...
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This note was uploaded on 02/21/2012 for the course MATH 4377 taught by Professor Staff during the Summer '08 term at University of Houston.
 Summer '08
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 Math

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