HW6s - bf MATH 6322, Complex Analysis Fall 2011, Key to...

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Unformatted text preview: 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: a ( z ) = z- a 1- az Prove that a). a maps the unit disc into unit disc, and a is holomorphic on D (0 , 1). b). Prove that a : D (0 , 1) (0 , 1) is one-to-one and onto and its inverse is - a . c). Use a), b) and Schwartz Lemma to prove the Schwartz-Pick theorem. Proof. a). It is clear that 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 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) < since | z | < 1 and | a | < 1. Hence, | z- a | 2- | 1- az | 2 , which implies z- a 1- az < 1, i.e. | a ( z ) | < 1 , for each z D (0 , 1) . b). For each w D (0 , 1), we solve a ( z ) = w , i.e. z- a 1- az = w, 1 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), a is one-to-one and onto, and the inverse of a ( z ) is - a ( w ) = w + a 1 + aw , w D (0 , 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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HW6s - bf MATH 6322, Complex Analysis Fall 2011, Key to...

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