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math185f09-hw8 - 1 5 Let S = x iy ∈ C | x,y ∈[0 1 be...

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MATH 185: COMPLEX ANALYSIS FALL 2009/10 PROBLEM SET 8 1. Let Ω be a region containing D (0 , 1). Let f : Ω C be analytic. (a) Let M > 0 be a constant. Suppose | f ( z ) | ≥ M for all z ∂D (0 , 1) and | f (0) | < M . Show that f has at least one zero in D (0 , 1). (b) Suppose | f ( z 2 ) | ≥ | f ( z ) | for all z D (0 , 1). Show that f is constant on D (0 , 1). 2. Let f and g be functions analytic on a bounded region Ω and continuous on Ω. Prove that | f | + | g | attains its maximum value on the boundary of Ω, i.e. max z Ω | f ( z ) | + | g ( z ) | = max z Ω | f ( z ) | + | g ( z ) | . 3. Suppose g 0 ( z ) 6 = 0 for all z Ω. Show that { Re g ( z ) + Im g ( z ) R | z Ω } is an open subset of R . 4. Let a C and | a | ≤ 1. Consider the polynomial P ( z ) = a 2 + (1 - | a | 2 ) z - a 2 z 2 . Show that | P ( z ) | ≤ 1 whenever | z | ≤
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Unformatted text preview: 1. 5. Let S = { x + iy ∈ C | x,y ∈ [0 , 1] } be the unit square in C . Let f be analytic on a region Ω that contains S . Suppose the following is true: (i) for all z with 0 ≤ Re( z ) ≤ 1, and Im( z ) = 0 or 1, Re f ( z ) = 0; (ii) for all z with Re( z ) = 0 or 1, and 0 ≤ Im( z ) ≤ 1, Im f ( z ) = 0 . Show that f ≡ 0 on Ω. [ Hint : Consider the function e-if ( z ) 2 . Apply both Maximum Modulus Theorem and Open Mapping Theorem.] Date : November 14, 2009 (Version 1.0); due: November 20, 2009. 1...
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