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Unformatted text preview: Serge Ballif MATH 502 Homework 6 February 29, 2008 (1) The function f ( z ) is entire, and it is known that  f ( z )  ≤ 1 whenever  z  = 1 and that  f ( z )  ≤ 10 whenever  z  = 10 . Show that  f ( z )  ≤ 5 whenever  z  = 5 . What can be said about  f (0)  ? Claim 1. For all 1 ≤  z  ≤ 10 we have  f ( z )  ≤  z  . Proof. Consider the function f ( z ) /z defined on A = { 1 <  z  < 10 } . This region is a connected open set on which f ( z ) /z is holomorphic (since no poles lie in the domain). Hence, we can invoke the maximum principle to see that  f ( z ) /z  attains its maximum on the the boundary ∂A = { z  = 1 , 10 } . By assumption we know that  f ( z ) /z  ≤ 1 on ∂A . Thus, for every z ∈ A , we must have  f ( z ) /z  ≤ 1 or equivalently  f ( z )  ≤  z  . In particular, this shows that  f ( z )  < 5 for  z  = 5. The only thing that we can conclude about f (0) is that it lies somewhere in the unit disc, because for any u ∈...
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 Summer '08
 JOHNRIDENER
 Math, TA, Holomorphic function, Types of functions, Functions and mappings, Conformal map

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