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math185f09-hw9 - MATH 185 COMPLEX ANALYSIS FALL 2009/10...

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MATH 185: COMPLEX ANALYSIS FALL 2009/10 PROBLEM SET 9 1. Consider the functions defined by g a ( z ) = e iπz/ 2 - 1 e iπz/ 2 + 1 and g b ( z ) = e πz/ 2 - 1 e πz/ 2 + 1 . Show that g a maps the set Ω a := { z C | - 1 < Re z < 1 } to D (0 , 1) while g b maps the set Ω b := { z C | - 1 < Im z < 1 } to D (0 , 1). Hence or otherwise, prove the following. (a) Let f : D (0 , 1) C be an analytic function that satisfies f (0) = 0. Suppose | Re f ( z ) | < 1 for all z D (0 , 1). By considering the function g a f or otherwise, prove that | f 0 (0) | ≤ 4 π . (b) Let S be the set of functions defined by S = { f : Ω b C | f analytic, | f | < 1 on Ω b , and f (0) = 0 } . By considering the function f g - 1 b or otherwise, Prove that sup f ∈S | f (1) | = e π/ 2 - 1 e π/ 2 + 1 . 2. Let f : D (0 , 1) C be analytic and | f ( z ) | < 1 for all z D (0 , 1). Recall that ϕ α ( z ) = ( z - α ) / (1 - αz ). (a) By considering the function ϕ f ( a ) f ϕ - a or otherwise, show that | f 0 ( z ) | 1 - | f ( z ) | 2 1 1 - | z | 2 . (b) Suppose there exist two distinct points a, b D (0 , 1) such that f ( a ) = a and f ( b ) = b . By considering the function ϕ a f ϕ - a
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