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Math Help - complex number

  1. #1
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    complex number

    If z is a non-zero complex number, prove that
    |conjugate(z) + 1/z| >=2
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: complex number

    |\bar{z}+1/z|\geq 2\Leftrightarrow |\bar{z}+1/z|^2\geq 4 . Using |w|^2=\bar{w}w we obtain

    |\bar{z}+1/z|^2=\ldots =|z|^2+2+1/|z|^2\geq \ldots \geq 4 .
    Last edited by FernandoRevilla; July 15th 2011 at 09:00 AM.
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  3. #3
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    Re: complex number

    Would you please derive more detail why modulus(z)^2 + 2 + 1/modulus(z)^2 >= 4?
    Many thanks.
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  4. #4
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    Re: complex number

    Quote Originally Posted by tinng View Post
    Would you please derive more detail why modulus(z)^2 + 2 + 1/modulus(z)^2 >= 4?
    Many thanks.
    What does the graph of y = x^2 + 2 + \frac{1}{x^2} look like ....? What is its range ....? (Have you learned calculus and therefore know how to find coordinates of turning points?)
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  5. #5
    MHF Contributor FernandoRevilla's Avatar
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    Re: complex number

    Quote Originally Posted by tinng View Post
    Would you please derive more detail why modulus(z)^2 + 2 + 1/modulus(z)^2 >= 4?
    Another way, prove that x^2+\dfrac{1}{x^2}\geq2\Leftrightarrow (x^2-1)^2\geq 0 . The last inequality is trivially satisfied.
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