Prove Inequality: 1/p + 1/q=1, u,v >= 0

[ehrenfest]

Homework Statement

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

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

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...

 

[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 don't 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