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Math Help - complex variables

  1. #1
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    complex variables

    Hi,

    I need two proofs for complex inequalities.

    1) |1-z|/z <2 or equal.

    2) |1+z|/|z|>1/2 or equal

    thanks
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  2. #2
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    Quote Originally Posted by Gilbert View Post
    Hi,

    I need two proofs for complex inequalities.

    1) |1-z|/z <2 or equal.

    2) |1+z|/|z|>1/2 or equal

    thanks

    Proofs...of what? I think you have to describe the set of points on the (complex) plane that fulfill the given inequalities, so put z=x+iy and (I think an absolute value (modulus) is lacking in the denominator of the first one):

    \frac{|1-z|}{|z|}\leq 2\Longrightarrow \frac{|(1-x)-yi|^2}{|x+iy|^2}\leq 4\Longrightarrow \frac{1-2x+x^2+y^2}{x^2+y^2}\leq 4 \Longrightarrow 3x^2 +3y^2+2x-1\geq 0\Longrightarrow 3\left(x+\frac{1}{3}\right)^2-\frac{1}{3}+3y^2-1\geq 0 \Longrightarrow \left(x+\frac{1}{3}\right)^2+y^2\geq \frac{4}{9}

    And we got a nice circle with center__ and radius__ . Now you try the second one.

    Tonio
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  3. #3
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    complex variables

    Hi,

    maybe not a proof,but I have to show these inequalities.I need them for

    the bound of a complex integral.

    I taught something like that

    |z+1|>|z-1|=1/2|z|+|1/2|z|-1|>1/2|z| but I'm not sure it's correct.

    Tanks
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  4. #4
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    Quote Originally Posted by Gilbert View Post
    Hi,

    maybe not a proof,but I have to show these inequalities.I need them for

    the bound of a complex integral.

    I taught something like that

    |z+1|>|z-1|=1/2|z|+|1/2|z|-1|>1/2|z| but I'm not sure it's correct.

    Tanks

    You cannot "show those inequalities" because they're not true for all complex numbers z...for example, check that inequality (1) is false when z=\frac{1}{2}i , and inequality (2) is false for example for z=-1

    Tonio
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  5. #5
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    Apr 2008
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    Complex variables

    Hi,

    yes you are right,but it's my fault.I have to show this inequalities for

    z->infinity
    as an upper bound and lower bound.

    Sorry

    Thanks
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