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Math Help - Cpx Analysis: Lower bound on a derivative.

  1. #1
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    Cpx Analysis: Lower bound on a derivative.

    The Problem:

    Suppose that \Omega is a simply connected domain with \Omega \ne \mathbb{C}, a \in \Omega, and for some r > 0, D(a;r) \subseteq \Omega (an open disk at a with radius r). For \mathbb{D} a unit disk, let f: \mathbb{D} \rightarrow \Omega be a conformal map with f(0) = a. Prove that |f'(0)| \ge r.

    Ideas:

    - There is a drawing (.pdf) attached that represents the set context.

    - Application of the Riemann Ext Thm. provides a conformal map h1 from \Omega onto \mathbb{D} with h1(a) = 0 and h1'(a) > 0 (or another suitable rotation) and a map h2 from f(\mathbb{D}) onto \mathbb{D}, h2(a) = 0 and h2'(a) > 0.

    - Seems like a setup for Schwarz's Lemma. For example, we can take g = h1 \circ f with g(0) = 0 and g(\mathbb{D}) \subseteq \mathbb{D}. Then the lemma asserts that |g(z)| \le |z| and |g'(0)| \le 1. So |h1'(f(0))| |f'(0)| \le 1 \Rightarrow |h1'(a)| |f'(0)| \le 1. I don't see where this helps with this proof.

    Thank you in advance for your help in solving this.
    Attached Thumbnails Attached Thumbnails Cpx Analysis: Lower bound on a derivative.-6-04-1-drawing.pdf  
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Hi Huram!

    Have you considered the map g(z)=\frac{1}{r}(z-a), which takes D(a,r) to \mathbb{D}?
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  3. #3
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    Solution

    Bruno: That was the clue that I needed. My rendition of a solution is attached. Thank you.
    Attached Thumbnails Attached Thumbnails Cpx Analysis: Lower bound on a derivative.-ex1_prelim_6_04.pdf  
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