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Inequality problem

  1. Oct 3, 2004 #1
    I would appreciate some ideas on the following problem:

    We are given the inequality:
    |a| + |b| >= 2(|c| + |d|)

    Can we conclude that:
    (c + a)^2 + (d + b)^2 >= c^2 + d^2 ?

    a, b, c, d are real.

    I can prove (2) if (1) is in the form: |a| + |b| >= 4max(|c|,|d|), but the above is stronger.

    Thank you.
  2. jcsd
  3. Oct 3, 2004 #2
    i feel its true
    its seems so easy but at 12.30 midnight i cannot think of anything!! :frown:

    -- AI
  4. Oct 4, 2004 #3


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    Science Advisor

    Yep I looked at it for a few minutes late last night and thought it looked easy but couldn't come up with much. I got another weaker result pretty easy but then decided to leave it for someone else.

    BTW, I got the weaker result by assumming (without loss of generality) that |b| >= |a|. I then found that if I made the additional imposition that |c|>=|d| then I was able to prove the desired result pretty easily . This however definitely does give a loss of generality and hence a weaker result.
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