HW6 - to prove the Schwarz-Pick Theorem (Theorem 2.5.2 on...

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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 unit-disc 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 one-to-one 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 Schwarz-Pick Theorem (Theorem 2.5.2 on Page 172). Extra 2 (Hint: Use Schwarz-Pick 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.

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