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Math Help - Complex variable inequality

  1. #1
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    Complex variable inequality

    Verify that  \sqrt[ ]{2}  \left |z   \right | \geq \left | Re(z)  \right | + \left | Im(z)   \right |

    Suggestion: reduce this inequality to {( \left | x  \right | - \left |  y \right |)}^{2}   \geq 0

    My steps:
    \sqrt[ ]{2} \left |z \right | \geq \left | x \right | + \left |y \right |

    2 {\left |z \right |}^{2} \geq {(\left | x \right | + \left |y \right |)}^{2}

    2 ({ x}^{2 } + { y}^{ 2}) \geq    {\left |  x \right |  }^{ 2} + 2 \left |  x \right |  \left | y  \right | +   { \left |  y \right |}^{2 }

    Where to go from here?
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  2. #2
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    Quote Originally Posted by davesface View Post
    Verify that  \sqrt[ ]{2}  \left |z   \right | \geq \left | Re(z)  \right | + \left | Im(z)   \right |

    Suggestion: reduce this inequality to {( \left | x  \right | - \left |  y \right |)}^{2}   \geq 0

    My steps:
    \sqrt[ ]{2} \left |z \right | \geq \left | x \right | + \left |y \right |

    2 {\left |z \right |}^{2} \geq {(\left | x \right | + \left |y \right |)}^{2}

    2 ({ x}^{2 } + { y}^{ 2}) \geq    {\left |  x \right |  }^{ 2} + 2 \left |  x \right |  \left | y  \right | +   { \left |  y \right |}^{2 }

    Where to go from here?
    Remember that x and y are real numbers, so that x^2 = |x|^2 and y^2 = |y|^2. If you then take everything over to the left side of that last inequality, it becomes \bigl(|x|-|y|\bigr)^2\geqslant0, which is obviously true.
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  3. #3
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    Wow, that was blindingly obvious. Thanks for the tip.
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