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Math Help - Upper and Lower Bounds

  1. #1
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    Upper and Lower Bounds

    Find an upper and lower bound for the reciprocal of the modulus of z^4-5z^2+6 if |z|=2

    n\leq\frac{1}{|z^4-5z^2+6|}\leq m

    Out of all the the ways to break modulus how do I know which structure of the triangle inequalities to use?

    |z^4-5z^2+6|=|(z^2-3)(z^2-2)|\geq|z^2-3||z^2-2|=||z|^2-|3||*||z^2|-|2|| =|4-3||4-2|=1*2=2

    |z^4-5z^2+6|\leq|z|^4-5|z|^2+6=2^4-5*2^2+6=16-20+6=2

    Why am I getting the same value for my upper and lower bounds?
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  2. #2
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    It is not necessary to do that. A is an upper bound for 1/f(x) if and only if 1/A is a lower bound (not 0) for f(x) and B is a lower bound for 1/f(x) if and only if 1/B is an upper bound for f(x).
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  3. #3
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    I don't understand since the bounds are \frac{1}{42}\leq\frac{1}{modulus}\leq\frac{1}{2}
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