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Prove an inequality

  • Thread starter ehrenfest
  • Start date
  • #1
2,012
1

Homework Statement


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]


Homework Equations





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...
 

Answers and Replies

  • #2
2,012
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It certainly works when u=v=1...
 
  • #3
45
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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.
 
  • #4
2,012
1
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]
Well, that comes from just multiplying the first equation by pq.



Then start to work on rearranging the second inequality, in order to remove the fractions.
So, I have been trying this and not getting anywhere. I tried plugging in p q = p+q. Can you be more specific?
 
  • #5
144
0
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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