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Help proving complex inequality

  1. Nov 8, 2011 #1
    This may seem trivial, but for some reason I am having trouble with it. For a and b in the complex plane, I am trying to prove the following:

    |a|^2+|b|^2 >= |(a+b)/2|^2

    I need this for part of a larger proof.
     
  2. jcsd
  3. Nov 9, 2011 #2
    Since noone has answered yet, I'll give it a go.

    Starting from the triangle inequality, we get

    |a+b| <= |a| + |b|

    =>

    |a+b|^2 <= (|a| + |b|)^2 = |a|^2 + |b|^2 + 2|a||b|

    =>

    |(a+b)/2|^2 <= |a|^2 / 4 + |b|^2 / 4 + |a||b| / 2


    If we can prove that |a|^2 + |b|^2 >= |a|^2 / 4 + |b|^2 / 4 + |a||b| / 2, then we're done!

    Can you go from here?
     
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