prove an inequality

[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...

[ehrenfest]

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

[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.

[ehrenfest]

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

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?

[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