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Math Help - Upperbound (Complex Analysis)

  1. #1
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    Upperbound (Complex Analysis)

    I am trying to figure out the following problem:
    Find an upper bound on |
    z3-2z2+6z+2| defined by |z|<2.

    I've done the following:
    |z3-2z2+6z+2|
    = |(x+iy)3-2(x+iy)2+6(x+iy)+2|
    = |x3+3ix2y-3xy2-iy3-2(x2+2ixy-y2)+6x+6iy+2|
    = |x3+3ix2y-3xy2-iy3-2x2-4ixy+2y2+6x+6iy+2|
    = |x3-3xy2-2x2+2y2+6x+2+(3x2y-y3-4xy+6y)i|

    = √(x3-3xy2-2x2+2y2+6x+2)2+ (3x2y-y3-4xy
    +6y)2

    I don't think I'm going this the right way. Any ideas? Should I not change to x+iy form?
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  2. #2
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    Re: Upperbound (Complex Analysis)

    by repeated applications of the triangle inequality

    $\left|z^3-2z^2+6z+2\right| \leq |z^3| + |2z^2|+|6z|+|2| = |z|^3 + 2 |z|^2 + 6|z| + 2$

    and if $|z|< 2$ then

    $\left|z^3-2z^2+6z+2\right| < 2^3 + 2\cdot 2^2 + 6\cdot 2 + 2 = 30$
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  3. #3
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    Re: Upperbound (Complex Analysis)

    I'm sorry, guys. I read the problem wrong. I need to use the triangle inequalities. I overcomplicated this problem.
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  4. #4
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    Re: Upperbound (Complex Analysis)

    Quote Originally Posted by romsek View Post
    by repeated applications of the triangle inequality

    $\left|z^3-2z^2+6z+2\right| \leq |z^3| + |2z^2|+|6z|+|2| = |z|^3 + 2 |z|^2 + 6|z| + 2$

    and if $|z|< 2$ then

    $\left|z^3-2z^2+6z+2\right| < 2^3 + 2\cdot 2^2 + 6\cdot 2 + 2 = 30$
    Thanks! I figured it out as soon as I finished typing it and submitting it. I over complicated things! Any way I can delete this post since it doesn't really contribute much to the community?
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