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Math Help - Analytic + bounded --> Uniformly continoius

  1. #1
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    Analytic + bounded --> Uniformly continoius

    Here is a problem I got stuck on.

    Let f be analytic and bounded in the half plane \{z:Rez>0\}. Show that for every c>0, f is uniformly continous in the half plane \{ z:Rez>c\}.

    I tried the following but got stuck.

    for every \epsilon >0 choose \delta = min(\frac{\epsilon}{M},c), where M is the bound of f. So for every |z_2-z_1|<\delta we get by Cauchys Integral formula |f(z_2)-f(z_1)|=\frac{1}{2\pi}|\int_{C_{\delta}} \frac{f(w)}{w-z_2} - \frac{f(w)}{w-z_1} dw|=\frac{1}{2\pi}|\int_{C_{\delta}} f(w)\frac{z_2-z_1}{(w-z_2)(w-z_1)} dw|\leq \frac{4*2\pi \delta}{2\pi \delta^2}M|z_1-z_2|\leq 2M. Which gets me nowhere.

    I'd appreciate some direction.
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  2. #2
    Member Abu-Khalil's Avatar
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    What is C_\delta?
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  3. #3
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    Quote Originally Posted by Abu-Khalil View Post
    What is C_\delta?
    Sorry I should have explained.

    It is a circle with radius \delta that containes both z_1,z_2.

    SK
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