Unformatted text preview: f : C → C be an entire function. (a) Suppose lim  z →∞ f ( z ) z = 0 . Show that f is a constant function. (b) Suppose f (0) = 3 + 4 i and  f ( z )  ≤ 5 for all z ∈ D (0 , 1). What is f (0)? 5. 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) Let Γ be the image of ∂D (0 , 1) under f . Show that Z Γ  dz  ≥ 2 π  f (0)  . Date : November 7, 2008 (Version 1.0); due: November 14, 2008. 1...
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 Fall '07
 Lim
 Math, Suppose

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