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HW6s - bf MATH 6322 Complex Analysis Fall 2011 Key to HW#6...

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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 one-to-one and onto and its inverse is φ - a . c). Use a), b) and Schwartz Lemma to prove the Schwartz-Pick 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 one-to-one 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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