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Math Help - help for question on bound of f(0)

  1. #1
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    help for question on bound of f(0)

    Let f(z) be a function which is analytic on and inside the circle |z| = 1 and which satisfies |f(z)|\leq{M} for all z on the circle |z| = 1. Prove that |f(0)|\leq{M}.

    By Cauchy's inequality i managed to obtain
    |f'(0)|\leq{\frac{1!M}{1^1}=M}.

    How do i proceed to prove that |f(0)|\leq{M}? Any help/suggestion is welcome!
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: help for question on bound of f(0)

    Quote Originally Posted by alphabeta89 View Post
    How do i proceed to prove that |f(0)|\leq{M}? Any help/suggestion is welcome!
    Hint: 0!=1 and 1^0=1.
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    Re: help for question on bound of f(0)

    use the Cauchy integral formula to evaluate and estimate f(0) as an integral over the unit circle.
    Last edited by hedi; November 16th 2012 at 04:16 AM.
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    Re: help for question on bound of f(0)

    Quote Originally Posted by FernandoRevilla View Post
    Hint: 0!=1 and 1^0=1.
    Hi, does Cauchy's inequality hold for |f^{(0)}(0)| in this case?
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    Re: help for question on bound of f(0)

    Yes,see attached.
    Attached Thumbnails Attached Thumbnails help for question on bound of f(0)-001.jpg  
    Thanks from alphabeta89
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    Re: help for question on bound of f(0)

    Quote Originally Posted by hedi View Post
    Yes,see attached.
    Thanks man!
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