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Let [itex]x \in \mathbb{R}^n[/itex] and

[tex]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[/tex]

Im having trouble proving the following inequality

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

I know i have to use Jensen's inequality

[tex] f\left(\frac{1}{|\Omega|}\int\limits_\Omega u dx \right) \le \frac{1}{|\Omega|}\int\limits_\Omega f(u) dx [/tex],

where [itex]f(u)[/itex] 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?

[tex]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[/tex]

Im having trouble proving the following inequality

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

I know i have to use Jensen's inequality

[tex] f\left(\frac{1}{|\Omega|}\int\limits_\Omega u dx \right) \le \frac{1}{|\Omega|}\int\limits_\Omega f(u) dx [/tex],

where [itex]f(u)[/itex] 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?

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