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Inequality problem

  1. Nov 30, 2006 #1
    Sorry that i posted in the wrong topic, i'm kind of new here :D
    Hi this is my problem:

    if 0<|z|<1 and z_1 = -1/a - ((1-a^2)^(1/2))/a
    z_2 = -1/a + ((1-a^2)^(1/2))/a
    Then it is clear to me that |z_1|>1 since using triangle inequality we get that |z_1| =| -1/a - ((1-a^2)^(1/2))/a | >= |1/a| + something smaller than one but positiv, and since |1/a| >1 then |z_1| > 1

    But how to prove |z_2| < 1 since bye triangle inequality we kind of get the same thing |z_2| = | -1/a + ((1-a^2)^(1/2))/a | >= |1/a|+ |((1-a^2)^(1/2))/a| > 1 ???? This doesnt make sense at all!

    Please help me, i need this to a problem on an integral in complex analysis, which i'm preparing for my exam ;)

    thank you for your time!
  2. jcsd
  3. Dec 1, 2006 #2


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    What exactly did you learn as "the triangle inequality"? Most people learn it as [itex]|a+ b|\le |a|+ |b|[/itex]. From that, if we let a= x- y, b= y we get
    [itex]|x-y+y|= |x|\le |x-y|+ |y|[/itex] so that [itex]|x-y|\ge |x|- |y|[/itex]. That second inequality is what you used. To prove the second part use the inequality [itex]|a+b|\le |a|+ |b|[/itex].
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