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Prove an inequality

  1. May 1, 2008 #1
    1. The problem statement, all variables and given/known data
    Let p and q be positive real numbers such that

    1/p + 1/q=1

    Prove that if [itex]u\geq 0[/itex] and [itex]v \geq 0[/itex], then

    [tex]uv \leq \frac{u^p}{p}+\frac{v^q}{q}[/tex]


    2. Relevant equations



    3. The attempt at a solution
    I am really stumped. Is there like a famous inequality that I need to use here that I am forgetting?
    This vaguely reminds me of the AM-GM inequality...
     
  2. jcsd
  3. May 2, 2008 #2
    It certainly works when u=v=1...
     
  4. May 2, 2008 #3
    I think I got it, but it was tricky.

    First, try to prove that [tex]\frac{1}{p}+\frac{1}{q}=1[/tex] can be rearranged to show that [tex]p+q=pq[/tex]

    Then start to work on rearranging the second inequality, in order to remove the fractions.
     
  5. May 2, 2008 #4
    Well, that comes from just multiplying the first equation by pq.



    So, I have been trying this and not getting anywhere. I tried plugging in p q = p+q. Can you be more specific?
     
  6. May 3, 2008 #5
    I dont know what known inequalities you are allowed to use, but I have a boring solution.
    There is a known generalization of the AM-GM inequality, called weighted AM-GM. check http://mathworld.wolfram.com/WeightedMean.html
     
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