homework108c5

homework108c5 - C → C is holomorphic Moreover assume...

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HOMEWORK 5 FOR MA108C: COMPLEX ANALYSIS DUE DATE: 4PM, MAY 12, 2009 (1) Let F : H C be holomorphic and satisfy | F ( z ) | ≤ 1 and F ( i ) = 0. Prove that | F ( z ) | ≤ ± ± ± ± z - i z + i ± ± ± ± , for all z H . Here H = { z C | Im ( z ) > 0 } . (2) Let P t ( z ) = n j =0 a j ( t ) z j with a j continuous in t . Show that if P t 0 has a simple zero at z 0 , then there exists ±,δ > 0 such that | t - t 0 | < δ = ⇒ ∃ ! z D ( z 0 ) : P t ( z ) = 0 . What happens if P t 0 vanishes at z 0 with order k ? (3) A fixed point of a function f : D D is a point z D such that f ( z ) = z . Prove that if f : D (0 , 1) D (0 , 1) is holomorphic and has two fixed points, then f is the identity; i.e. f ( z ) = z for all z D (0 , 1). (4) Suppose f :
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Unformatted text preview: C → C is holomorphic. Moreover assume there exist constants c 1 ,c 2 ,± > 0 such that | f ( z ) | ≤ c 1 exp( c 2 | z | 1 / 2-± ) for all z ∈ C , and that | f ( z ) | ≤ 1 for z ∈ R ≤ . Show that f is constant. Show that f ( z ) = cos( √-z ) (which is well-defined by evenness of cos) satisfies the conditions of the problem with ± = 0, but not the conclusion. Hint: Prove a Phragm´ en-Lindel¨of type theorem. 1...
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This document was uploaded on 12/08/2010.

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