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Thread: Complex variable inequality

  1. #1
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    Complex variable inequality

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

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

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

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

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

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

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

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

    $\displaystyle 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 $\displaystyle x^2 = |x|^2$ and $\displaystyle y^2 = |y|^2$. If you then take everything over to the left side of that last inequality, it becomes $\displaystyle \bigl(|x|-|y|\bigr)^2\geqslant0$, 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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