math185-hw7 - π/ 2-1 e π/ 2 + 1 . 3. Let f : D (0 , 1)...

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MATH 185: COMPLEX ANALYSIS FALL 2007/08 PROBLEM SET 7 You may use without proof any results that had been proved in the lectures. 1. Let f : C C be an entire function. Show that if | f ( z ) | ≤ 1 | Re z | for all z C , then f 0. What happens if we replace the condition by | f ( z ) | ≤ 1 | Im z | for all z C ? 2. 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), 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 } . Prove that sup f ∈S | f (1) | = e
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Unformatted text preview: π/ 2-1 e π/ 2 + 1 . 3. Let f : D (0 , 1) → C be analytic and | f ( z ) | < 1 for all z ∈ D (0 , 1). (a) Show that | f ( 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 . Show that f ( z ) = z for all z ∈ D (0 , 1). (c) Suppose there exist a ∈ D (0 , 1), a 6 = 0, such that f ( a ) = 0 = f (-a ). Show that | f (0) | ≤ | a | 2 . What can you conclude if | f (0) | = | a | 2 ? Date : November 8, 2007 (Version 1.2); posted: November 8, 2007; due: November 14, 2007. 1...
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This note was uploaded on 08/01/2008 for the course MATH 185 taught by Professor Lim during the Fall '07 term at University of California, Berkeley.

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