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Math Help - Analytic problem

  1. #1
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    Analytic problem

    Let  f(z) be analytic in the disk |z| <1. If f(z) has a zero of order 2 at the origin and |f(z)| \le 1 in that disk. Prove that |f(z)|\le|z|^2 in |z|<1

    I have no idea where to start. Thank you.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Analytic problem

    Hint: In the series expansion f(z)=\sum_{n=0}^{+\infty}a_nz^n we have a_0=a_1=0 , so f(z)/z^2 is analytic in |z|<1 .
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  3. #3
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    Re: Analytic problem

    Thank you for you reply, I could do what you said but whats the next step? Thanks
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Re: Analytic problem

    By hypothesis |f(z)|\leq 1 so, |f(z)/z^2|\leq 1/r^2 for |z|=r . This equality is also valid for |z|\leq r according to the Maximum Modulus Principle. If we fix z in |z|<1 we have |f(z)|\leq |z|^2/r^2 for all r\geq |z| and <1 . You can conclude.
    Last edited by FernandoRevilla; December 16th 2011 at 06:59 AM.
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    Re: Analytic problem

    Quote Originally Posted by FernandoRevilla View Post
    By hypothesis |f(z)|\leq 1 so, |f(z)/z^2|\leq 1/r^2 for |z|=r . This equality is also valid for |z|\leq r according to the Maximum Modulus Principle. If we fix z in |z|<1 we have |f(z)|\leq |z|^2/r^2 for all r\geq |z| and <1 . You can conclude.
    Sorry but I didn't get it. How can I deduce |f(z)/z^2|\leq 1/r^2for |z|\leq r according to the Maximum Modulus Principle? Thanks!!!
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  6. #6
    MHF Contributor FernandoRevilla's Avatar
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    Re: Analytic problem

    Quote Originally Posted by fareastmovement View Post
    Sorry but I didn't get it. How can I deduce |f(z)/z^2|\leq 1/r^2for |z|\leq r according to the Maximum Modulus Principle? Thanks!!!
    What does the Maximum Modulus Principle say?
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  7. #7
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    Re: Analytic problem

    Suppose f is analytic on |z-z_0|<\epsilon. If |f(z)| \le |f(z_0)| for z on this region then f(z) \equiv f(z_0) on the region.

    but |f(z)/z^2|\leq 1/r^2. How can I make sure that f(z_0)=1/r^2? Or I am in the wrong direction? Thank you
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  8. #8
    MHF Contributor FernandoRevilla's Avatar
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    Re: Analytic problem

    Better use this version: Let D \subset \mathbb{C} be a bounded domain, and let f be a continuous function on the closed set \overline{D} that is analytic on D. Then the maximum value of |f| on \overline{D} (which always exists) occurs on the boundary \partial D . So, in our case, it is not possible |f(z)/z^2|>1/r^2 if |z|<r .
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