math185f08-hw8

math185f08-hw8 - f α = 0(a Let Ω = H a and α = 1 Show...

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MATH 185: COMPLEX ANALYSIS FALL 2008/09 PROBLEM SET 8 For z C × , recall that the principle argument of z , denoted Arg( z ), is the unique θ [ - π,π ) such that z = | z | e . By convention Arg(0) = 0. 1. Let S denote the sector given by { z C | - π/ 4 < Arg( z ) < π/ 4 } . Let f : ¯ S C be a continuous function such that f is analytic on S . Suppose (i) | f ( z ) | ≤ 1 for all z ∂S ; (ii) | f ( x + iy ) | ≤ e x for all x + iy S . Prove that | f ( z ) | ≤ 1 for all z S . 2. (a) Show that the functions η a and η b maps D (0 , 1) to H a and H b where η a ( z ) = 1 + z 1 - z , η b ( z ) = i ± 1 + z 1 - z ² , and H a = { z C | Re z > 0 } , H b = { z C | Im z > 0 } . (b) Show that the functions σ a and σ b maps S a and S b to D (0 , 1) where σ a ( z ) = e iπz/ 2 - 1 e iπz/ 2 + 1 , σ b ( z ) = e πz/ 2 - 1 e πz/ 2 + 1 and S a = { z C | - 1 < Re z < 1 } , S b = { z C | - 1 < Im z < 1 } . 3. For a region Ω C and a point α Ω, let F ) be the set of functions defined by F ) := { f : Ω C | f analytic, | f | < 1 on Ω, and
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Unformatted text preview: f ( α ) = 0 } . (a) Let Ω = H a and α = 1. Show that sup f ∈F ( H a , 1) | f (1) | = 1 2 . (b) Let Ω = H b and α = i . Show that sup f ∈F ( H b ,i ) | f (2 i ) | = 1 3 . (c) Let Ω = S b and α = 0. Show that sup f ∈F ( S b , 0) | f (1) | = e π/ 2-1 e π/ 2 + 1 . 4. 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 . Date : November 14, 2008 (Version 1.1); due: November 21, 2008. 1 (b) Suppose f (0) = 0. Show that | f ( z ) + f (-z ) | ≤ 2 | z | 2 . 2...
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math185f08-hw8 - f α = 0(a Let Ω = H a and α = 1 Show...

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