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Need help with an inequality.

  1. Jun 16, 2007 #1
    Can anyone help me to prove this inequality:

    (a^2)/b+(b^2)/c+(c^2)/a >= a+b+c.

    I know i must use Cauchy formula, but can not prove it.

    Thank you .
     
  2. jcsd
  3. Jun 17, 2007 #2

    malawi_glenn

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    What do you mean, that you dont know how to prove cauchy's inequality? Or that you can not come to that point where you have the equation on the form of cauchy's inequality?
     
  4. Jun 17, 2007 #3
    Thank you. I've solved it myself. This is one of the homework series that i have to use cauchy's inequality.

    Can you help me to prove this one:

    a/(a+b)+b/(b+c)+c/(c+a) > sqrt(a/(b+c)) + sqrt(b/(a+c)) + sqrt(c/(a+b)).
     
  5. Jun 17, 2007 #4

    malawi_glenn

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    what have you done so far?..
     
  6. Jun 17, 2007 #5
    My idea by now is :
    a/(a+b)<1, b/(b+c)< 1 and c/(c+a)<1, so the left hand side is smaller than sqrt(a/(a+b))+sqrt(b/(b+c))+sqrt(c/(c+a)).
    Then I try to compare the right hand side with sqrt(a/(a+b))+sqrt(b/(b+c))+sqrt(c/(c+a)) because they are both in the square root type (i hope it will be easier). But it doesn't work so far. Can you give me some hints.
     
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