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Math Help - Complex Variables - Upper Bound

  1. #1
    Rhymes with Orange Chris L T521's Avatar
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    Complex Variables - Upper Bound

    I'm starting to work on this one question, but I'm not sure how to apply this inequality in it:

    \left|\left|z_1\right|-\left|z_2\right|\right|\leq\left|z_1+z_2\right|\le  q\left|z_1\right|+\left|z_2\right|

    The question is : Find an upper bound for \left|\frac{-4}{z^3-5z+1}\right| if \left|z\right|=2.

    Now, I managed (in a previous question) to show that the lower and upper bounds on \left|z^3-5z+1\right| when \left|z\right|=2 and I got 3\leq\left|z^3-5z+1\right|\leq 19....I have a feeling I need to incorporate this, but I'm not sure how.

    Any input would be appreciated!!
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  2. #2
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    Quote Originally Posted by Chris L T521 View Post
    I'm starting to work on this one question, but I'm not sure how to apply this inequality in it:

    \left|\left|z_1\right|-\left|z_2\right|\right|\leq\left|z_1+z_2\right|\le  q\left|z_1\right|+\left|z_2\right|

    The question is : Find an upper bound for \left|\frac{-4}{z^3-5z+1}\right| if \left|z\right|=2.

    Now, I managed (in a previous question) to show that the lower and upper bounds on \left|z^3-5z+1\right| when \left|z\right|=2 and I got 3\leq\left|z^3-5z+1\right|\leq 19....I have a feeling I need to incorporate this, but I'm not sure how.

    Any input would be appreciated!!
    I wonder how you got the bounds on |z^3-5z+1|: consider z=2... The best lower bound is |z^3-5z+1|\geq |5z|-|z^3+1|\geq 5|z|-|z|^3-1=1.

    Then all the work is done: \left|\frac{-4}{z^3-5z+1}\right|=\frac{4}{|z^3-5z+1|}\leq \frac{4}{1}=4 by the previous lower bound...
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  3. #3
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Laurent View Post
    I wonder how you got the bounds on |z^3-5z+1|: consider z=2... The best lower bound is |z^3-5z+1|\geq |5z|-|z^3+1|\geq 5|z|-|z|^3-1=1.
    To satisfy your curiosity, this is how I got the bounds:

    L.B: \left|z^3-5z+1\right|=\left|z^3-\left(5z-1\right)\right|\geqslant\left|\left|z^3\right|-\left|5z-1\right|\right|\geqslant\left|\left|z\right|^3-\left(5\left|z\right|+1\right)\right|=\left|-3\right|=3

    U.B: \left|z^3-5z+1\right|=\left|\left(z^3-5z\right)+1\right|\leqslant\left|\left|z^3-5z\right|+1\right|\leqslant\left|\left|z\right|^3+  5\left|z\right|+1\right|=19

    Then all the work is done: \left|\frac{-4}{z^3-5z+1}\right|=\frac{4}{|z^3-5z+1|}\leq \frac{4}{1}=4 by the previous lower bound...
    That makes sense. I figured that I had to do something like this.

    Thank you!
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  4. #4
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    Quote Originally Posted by Chris L T521 View Post
    To satisfy your curiosity, this is how I got the bounds:

    L.B: \left|z^3-5z+1\right|=\left|z^3-\left(5z-1\right)\right|\geqslant\left|\left|z^3\right|-\left|5z-1\right|\right|{\color{red}\geqslant}\left|\left|z  \right|^3-\left(5\left|z\right|+1\right)\right|=\left|-3\right|=3
    I feel like my previous remark about z=2 was too allusive. I was pointing at a mistake: it lies in the inequality in red. Indeed, the function x\mapsto|x| is not increasing, so you can't write what you did: \left|z^3\right|-\left|5z-1\right|\leq |z|^3-5|z|+1 does not imply \left|\left|z^3\right|-\left|5z-1\right|\right|\leq \left||z|^3-5|z|+1\right|.
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