# Using Jensen's Inequality

1. Apr 4, 2006

### ReyChiquito

Let $x \in \mathbb{R}^n$ and

$$u_0>0, \qquad \int\limits_\Omega u_0(x) dx =1, \qquad E(t)=\int\limits_\Omega u(x,t)u_0(x)dx$$

Im having trouble proving the following inequality

$$\int\limits_\Omega \frac{u_0(x)}{(1+u(x,t))^2}dx \ge \dfrac{1}{(1+E)^2}. \qquad \hbox{(1)}$$

I know i have to use Jensen's inequality

$$f\left(\frac{1}{|\Omega|}\int\limits_\Omega u dx \right) \le \frac{1}{|\Omega|}\int\limits_\Omega f(u) dx$$,

where $f(u)$ is convex.

But in order to use it to prove (1), I need to rewrite the left hand side of the equation or use a previous inequality right?

There is where im stuck. Can anybody give me a sugestion pls?

Last edited: Apr 4, 2006
2. Apr 4, 2006

### ReyChiquito

Is it just me or nobody can see the TeX?

3. Apr 4, 2006

### cogito²

I cannot either.

4. Apr 6, 2006

### ReyChiquito

Well, first of all, it would be nice if someone tell me why the TeX doesnt work. Second of all, i got it, so nevermind.

$$\int$$

$$\Omega$$

$$\omega$$

no \int???? nice.....

Last edited: Apr 6, 2006