Prove an inequality

1. May 1, 2008

ehrenfest

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 $u\geq 0$ and $v \geq 0$, then

$$uv \leq \frac{u^p}{p}+\frac{v^q}{q}$$

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. May 2, 2008

ehrenfest

It certainly works when u=v=1...

3. May 2, 2008

CrazyIvan

I think I got it, but it was tricky.

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

Then start to work on rearranging the second inequality, in order to remove the fractions.

4. May 2, 2008

ehrenfest

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?

5. May 3, 2008

Kurret

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