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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 onetoone and onto and its inverse is  a . c). Use a), b) and Schwartz Lemma to prove the SchwartzPick 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 &lt; az +  a  2 (1 2 &lt; az +  a  2  z  2 ) =  z  2 +  a  2 1  a  2  z  2 = (1  z  2 )(  a  2 1) &lt; since  z  &lt; 1 and  a  &lt; 1. Hence,  z a  2  1 az  2 , which implies z a 1 az &lt; 1, i.e.  a ( z )  &lt; 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  &lt; 1. Since z is unique for each w D (0 , 1), a is onetoone 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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