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Math Help - 2 problems in complex variable

  1. #1
    Junior Member
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    2 problems in complex variable

    Welcome my dear :

    I need your help in solving these 2 problems in complex variable. You will find my
    questions in the file attached.

    Please give me an idea and I will try to proceed.

    Thank you in advance
    Attached Files Attached Files
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  2. #2
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    I get the proof for the second question and what I want know is just the proof of Cauchy's inequality by induction
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by fuzzy topology View Post
    Welcome my dear :

    I need your help in solving these 2 problems in complex variable. You will find my
    questions in the file attached.

    Please give me an idea and I will try to proceed.

    Thank you in advance
    It's clear for the first case. Now suppose that \left|\sum_{i=1}^n a_ib_i\right|^2\leqslant \sum_{i=1}^n|a_i|^2\sum_{i=1}^n|b_i|^2 then \left|\sum_{i=1}^{n+1}a_ib_i\right|^2=\left|\sum_{  i=1}^{n}a_ib_i+a_nb_n\right|^2\leqslant\cdots.

    Finish.
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  4. #4
    Super Member Random Variable's Avatar
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    You probably have alread realized this but 1 + \cos \theta + \cos 2  \theta + \cos 3 \theta + ... + \cos n \theta= \frac{1}{2} + \frac{1}{2} \sum_{k=-n}^{n} e^{ik \theta}
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  5. #5
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    Thank you Drexel28 very much, I know understand the concept.
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