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

• ehrenfest
In summary, the given problem can be solved by using the weighted AM-GM inequality and rearranging equations to show that p+q=pq. Then, the second inequality can be simplified by removing the fractions.

## 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

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

## 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...

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

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.

CrazyIvan said:
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.

CrazyIvan said:
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?